From b0af8c6613adb7ba3ead8535e271b98f838bdb7a Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Thu, 14 Nov 2024 13:21:01 +0100 Subject: [PATCH 01/41] Fix up for ideal sheaves. --- src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl b/src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl index bba517f1af49..3ac9c2c3641f 100644 --- a/src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl +++ b/src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl @@ -757,7 +757,7 @@ is_locally_prime(I::PrimeIdealSheafFromChart) = true function is_equidimensional(I::AbsIdealSheaf; covering=default_covering(scheme(I))) has_attribute(I, :is_equidimensional) && return get_attribute(I, :is_equidimensional)::Bool - has_attribute(I, :is_prime) && return get_attribute(I, :is_prime)::Bool + has_attribute(I, :is_prime) && get_attribute(I, :is_prime)::Bool && return true local_dims = [dim(I(U)) for U in patches(covering) if !isone(I(U))] length(local_dims) == 0 && return true # This only happens if I == OO(X) d = first(local_dims) From 6e4cf759db4a519d023668003a1f51ec427b2cc2 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Fri, 15 Nov 2024 15:16:59 +0100 Subject: [PATCH 02/41] Widen some signatures. --- src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl | 2 +- src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl | 7 +++---- 2 files changed, 4 insertions(+), 5 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl index 82806d80c191..79660467c47f 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl @@ -395,7 +395,7 @@ end # Note that we need one minimal concrete type for the return values, # so the implementation can not be truly generic. -function +(D::T, E::T) where {T<:AbsAlgebraicCycle} +function +(D::AbsAlgebraicCycle, E::AbsAlgebraicCycle) X = ambient_scheme(D) X === ambient_scheme(E) || error("divisors do not live on the same scheme") R = coefficient_ring(D) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl index 34c5e1255791..da480dc40c9b 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl @@ -322,17 +322,16 @@ function weil_divisor(C::CartierDivisor) end @doc raw""" - intersect(W::WeilDivisor, C::EffectiveCartierDivisor; check::Bool=true) + intersect(W::AbsWeilDivisor, C::EffectiveCartierDivisor; check::Bool=true) Computes the intersection of ``W`` and ``C`` as in [Ful98](@cite) and returns an `AbsAlgebraicCycle` of codimension ``2``. """ -function intersect(W::WeilDivisor, C::EffectiveCartierDivisor; check::Bool=true) +function intersect(W::AbsWeilDivisor, C::EffectiveCartierDivisor; check::Bool=true) X = ambient_scheme(W) result = zero(W) for I in components(irreducible_decomposition(W)) inc_Y = CoveredClosedEmbedding(X, I, check=false) - #inc_Y = CoveredClosedEmbedding(X, I, covering=trivializing_covering(C), check=false) Y = domain(inc_Y) pbC = pullback(inc_Y)(C) # Will complain if the defining equation of C is vanishing identically on Y W_sub = weil_divisor(pbC) @@ -342,7 +341,7 @@ function intersect(W::WeilDivisor, C::EffectiveCartierDivisor; check::Bool=true) end @doc raw""" - intersect(W::WeilDivisor, C::CartierDivisor; check::Bool=true) + intersect(W::AbsWeilDivisor, C::CartierDivisor; check::Bool=true) Computes the intersection of ``W`` and ``C`` as in [Ful98](@cite) and returns an `AbsAlgebraicCycle` of codimension ``2``. From 63e383483d445100f374a331cbb4765ef112a42f Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Sun, 17 Nov 2024 19:13:31 +0100 Subject: [PATCH 03/41] Initialize experimental folder for contributions from Durham. --- experimental/Durham2024/README.md | 12 ++++++++++++ experimental/Durham2024/docs/doc.main | 7 +++++++ experimental/Durham2024/src/Durham2024.jl | 12 ++++++++++++ experimental/Durham2024/test/runtests.jl | 4 ++++ experimental/Experimental.jl | 1 + 5 files changed, 36 insertions(+) create mode 100644 experimental/Durham2024/README.md create mode 100644 experimental/Durham2024/docs/doc.main create mode 100644 experimental/Durham2024/src/Durham2024.jl create mode 100644 experimental/Durham2024/test/runtests.jl diff --git a/experimental/Durham2024/README.md b/experimental/Durham2024/README.md new file mode 100644 index 000000000000..564c57c64cc7 --- /dev/null +++ b/experimental/Durham2024/README.md @@ -0,0 +1,12 @@ +# Contributions to OSCAR from the Computational Algebraic Geometry workshop in Durham, 2024 + +## Aims + +This folder contains contributions made by participants of the workshop +on Computational Algebraic Geometry, held in Durham, November 18-22nd, 2024. + +## Status + +This is work in progress which will be updated throughout the week and eventually +cleaned up and finalized for a proper pull request to OSCAR. + diff --git a/experimental/Durham2024/docs/doc.main b/experimental/Durham2024/docs/doc.main new file mode 100644 index 000000000000..6bc6bb7bd30c --- /dev/null +++ b/experimental/Durham2024/docs/doc.main @@ -0,0 +1,7 @@ +# Additional functionality for intersection theory on schemes + +## Pushforward on algebraic cycles along proper morphisms +```@docs + pushforward(p::BlowupMorphism, W::AbsAlgebraicCycle) +``` + diff --git a/experimental/Durham2024/src/Durham2024.jl b/experimental/Durham2024/src/Durham2024.jl new file mode 100644 index 000000000000..f71df7f5e83e --- /dev/null +++ b/experimental/Durham2024/src/Durham2024.jl @@ -0,0 +1,12 @@ +# pushforward of algebraic cycles for proper maps + +@doc raw""" + pushforward(p::BlowupMorphism, W::AbsAlgebraicCycle) + +Given a `BlowupMorphism` ``p : X → Y`` and an `AbsAlgebraicCycle` ``W ∈ Z_k(X)``, +compute ``p_*(W) ∈ Z_k(Y)``. +""" +function pushforward(p::BlowupMorphism, W::AbsAlgebraicCycle) + error("not implemented") +end + diff --git a/experimental/Durham2024/test/runtests.jl b/experimental/Durham2024/test/runtests.jl new file mode 100644 index 000000000000..85a7ef3f1fa6 --- /dev/null +++ b/experimental/Durham2024/test/runtests.jl @@ -0,0 +1,4 @@ +@testset "Experimental template tests" begin + S = Oscar.ExampleStruct(5) + @test 5 == Oscar.my_access_func(S) +end diff --git a/experimental/Experimental.jl b/experimental/Experimental.jl index 878a6e356922..6b418b620d83 100644 --- a/experimental/Experimental.jl +++ b/experimental/Experimental.jl @@ -13,6 +13,7 @@ const orderedpkgs = [ "Schemes", "FTheoryTools", # must be loaded after Schemes and LieAlgebras "IntersectionTheory", # must be loaded after Schemes + "Durham2024", ] const exppkgs = filter(x->isdir(joinpath(expdir, x)) && !(x in orderedpkgs), readdir(expdir)) append!(exppkgs, orderedpkgs) From 0d1e98b216edbaf19514cc944990ee161d69a623 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Sun, 17 Nov 2024 19:14:50 +0100 Subject: [PATCH 04/41] Add first introductory notebook. --- .../Durham2024/material/Durham_2024_I.ipynb | 641 ++++++++++++++++++ 1 file changed, 641 insertions(+) create mode 100644 experimental/Durham2024/material/Durham_2024_I.ipynb diff --git a/experimental/Durham2024/material/Durham_2024_I.ipynb b/experimental/Durham2024/material/Durham_2024_I.ipynb new file mode 100644 index 000000000000..ce20d1867b70 --- /dev/null +++ b/experimental/Durham2024/material/Durham_2024_I.ipynb @@ -0,0 +1,641 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "8da163a2", + "metadata": {}, + "source": [ + "# Algebraic Geometry in OSCAR: part I\n", + "\n", + "[OSCAR](https://www.computeralgebra.de) is a computer algebra system whose development is funded by the Deutsche Forschungsgemeinschaft (DFG) as part of the SFB-TRR 195. It is open source and can be found on [GitHub](www.github.com/oscar-system/Oscar.jl). The core functionality is for all kinds of *symbolic computations* and comprises number theory, group and representation theory, polyhedral geometry, commutative algebra, ...\n", + "\n", + "Today, we will have a look at how to use OSCAR to do Algebraic Geometry and, in particular, Intersection Theory. We will mostly follow **[Ful98]** and the [stacks project](https://stacks.math.columbia.edu/)." + ] + }, + { + "cell_type": "markdown", + "id": "a55e6e76", + "metadata": {}, + "source": [ + "### Installation and starting up OSCAR \n", + "\n", + "Installation instructions for OSCAR can be found on its [website](https://www.oscar-system.org/install/). \n", + "For the purpose of this course, we shall need a very particular branch of OSCAR which can be installed using the following commands." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "aa985b7c", + "metadata": {}, + "outputs": [], + "source": [ + "using Pkg\n", + "]add https://github.com/HechtiDerLachs/Oscar.jl#Durham_2024" + ] + }, + { + "cell_type": "markdown", + "id": "dede3a91", + "metadata": {}, + "source": [ + "To use OSCAR, start up `julia` and type" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8aa0970f", + "metadata": {}, + "outputs": [], + "source": [ + "#using Revise, Oscar # Revise is another very useful, independent package \n", + " # which allows you to change code without restarting.\n", + "using Oscar" + ] + }, + { + "cell_type": "markdown", + "id": "5d5d4ffd", + "metadata": {}, + "source": [ + "You may need to do some updates, eventually:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f7c0f6e9", + "metadata": {}, + "outputs": [], + "source": [ + "]up" + ] + }, + { + "cell_type": "markdown", + "id": "21fba4ed", + "metadata": {}, + "source": [ + "### First steps for Algebraic Geometry in OSCAR\n", + "\n", + "Let us define an elliptic curve $E$ over the function field $\\mathbb k(t)$ for $\\mathbb k = \\mathbb F_{113}$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2c260e95", + "metadata": {}, + "outputs": [], + "source": [ + "kk = GF(113) # the finite field (Galois field) with 113 elements" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "4327a12e", + "metadata": {}, + "outputs": [], + "source": [ + "kkt_poly, t_poly = kk[:t] # the polynomial ring in one variable `t` over `kk`" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "159c11a4", + "metadata": {}, + "outputs": [], + "source": [ + "kkt_field = fraction_field(kkt_poly) # the fraction field of that polynomial ring" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0aa993cc", + "metadata": {}, + "outputs": [], + "source": [ + "t_field = kkt_field(t_poly) # the monomial `t`, but as an element of the fraction field" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "14367677", + "metadata": {}, + "outputs": [], + "source": [ + "P, (x, y) = kkt_field[:x, :y]; # the bivariate polynomial ring over `kkt_field`\n", + "P" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0135afd3", + "metadata": {}, + "outputs": [], + "source": [ + "f = y^2 - x^3 - (t_poly^8 - 1)*x # a Weierstrass equation for an elliptic curve" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "549c5e73", + "metadata": {}, + "outputs": [], + "source": [ + "E = elliptic_curve(f, x, y) # the associated elliptic curve as an instance of `EllipticCurve`" + ] + }, + { + "cell_type": "markdown", + "id": "c6ca76fa", + "metadata": {}, + "source": [ + "### Affine schemes in OSCAR\n", + "\n", + "For an elliptic curve as above with *integral* coefficients, we can build the *Weierstrass model* of its associated *elliptic surface*." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "379dea9b", + "metadata": {}, + "outputs": [], + "source": [ + "P, (x, y, t) = kk[:x, :y, :t] # a multivariate polynomial ring in three variables\n", + "f = y^2 - x^3 - (t^8 - 1)*x # the defining equation for the Weierstrass chart\n", + "I = ideal(P, f) # a principal ideal in `P`" + ] + }, + { + "cell_type": "markdown", + "id": "bbf497c0", + "metadata": {}, + "source": [ + "To any finitely generated `kk`-algebra we can associate its [spectrum](https://stacks.math.columbia.edu/tag/01HR)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "4709c18a", + "metadata": {}, + "outputs": [], + "source": [ + "IA3 = spec(P) # the affine scheme associated to `P`\n", + "W, inc_W = sub(IA3, I) # the closed subscheme defined by `I` together with its `ClosedEmbedding`\n", + "W" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d770d898", + "metadata": {}, + "outputs": [], + "source": [ + "W_sing, inc_W_sing = singular_locus(W) # the singular locus of `W` with its `ClosedEmbedding`\n", + "W_sing" + ] + }, + { + "cell_type": "markdown", + "id": "2ce08ef2", + "metadata": {}, + "source": [ + "The second return value is a [closed embedding](https://stacks.math.columbia.edu/tag/01HJ) which has its own datatype and its own methods in OSCAR." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f505834e", + "metadata": {}, + "outputs": [], + "source": [ + "typeof(inc_W_sing)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "53e3e75d", + "metadata": {}, + "outputs": [], + "source": [ + "methodswith(ClosedEmbedding)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "cb207814", + "metadata": {}, + "outputs": [], + "source": [ + "I_sing = image_ideal(inc_W_sing) # the ideal defining `W_sing` as a closed subscheme of `W`" + ] + }, + { + "cell_type": "markdown", + "id": "17583d78", + "metadata": {}, + "source": [ + "We can get our hands on the distict points of the singular locus via a primary decomposition." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2a69a6f4", + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "I_sing_dec = primary_decomposition(I_sing) # eight distinct points at `(x, y, t) = (0, 0, \\zeta_8^k)`" + ] + }, + { + "cell_type": "markdown", + "id": "63ed9eb3", + "metadata": {}, + "source": [ + "In OSCAR we can work with polynomial rings and their quotients, but also with *localizations*." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "5bf7175e", + "metadata": {}, + "outputs": [], + "source": [ + "U = hypersurface_complement(W, t^8-1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2a6a96a1", + "metadata": {}, + "outputs": [], + "source": [ + "Sigma, _ = singular_locus(U)\n", + "is_empty(Sigma)" + ] + }, + { + "cell_type": "markdown", + "id": "55f93291", + "metadata": {}, + "source": [ + "### Covered schemes in OSCAR\n", + "\n", + "The above were examples of *affine schemes* and their morphisms. These are but wrappers of objects in commutative algebra. But in order to construct the actual elliptic surface associated to $E$, we need to resolve the singularities of the Weierstrass model $W$. \n", + "\n", + "This will lead to *covered schemes* which are glued from affine schemes as for example described [here](https://stacks.math.columbia.edu/tag/01JA)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8ccc870e", + "metadata": {}, + "outputs": [], + "source": [ + "P1 = I_sing_dec[1][1] # an ideal in `OO(W)`\n", + "pr1 = blow_up(W, P1) # the projection morphism `W1 -> W`" + ] + }, + { + "cell_type": "markdown", + "id": "b1661ec9", + "metadata": {}, + "source": [ + "We have just [blown up](https://stacks.math.columbia.edu/tag/052P) the affine scheme `W` along the ideal `P1`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "3b119926", + "metadata": {}, + "outputs": [], + "source": [ + "W1 = domain(pr1); # a `CoveredScheme` over `kk`\n", + "cov = default_covering(W1) # the initial `Covering` used to define the scheme" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8b77f6d5", + "metadata": {}, + "outputs": [], + "source": [ + "U = patches(cov)[1] # an affine scheme; the first chart of `W1`" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "cdfde006", + "metadata": {}, + "outputs": [], + "source": [ + "V = patches(cov)[2]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0f47df3a", + "metadata": {}, + "outputs": [], + "source": [ + "glue = cov[U, V] # either two `patches` in a `Covering` might be glued along open subsets" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "872dc99d", + "metadata": {}, + "outputs": [], + "source": [ + "U_simp = simplify(U) # one can always try to eliminate variables using `simplify`" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "64edc87a", + "metadata": {}, + "outputs": [], + "source": [ + "E1 = exceptional_divisor(pr1) # the exceptional divisor of the blowup\n", + "typeof(E1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "cede2895", + "metadata": {}, + "outputs": [], + "source": [ + "E1(U_simp) # the defining equations for the exceptional divisor in `U_simp`" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "faf7c210", + "metadata": {}, + "outputs": [], + "source": [ + "small_generating_set(ideal_sheaf(E1)(U)) # a glimpse of the ideal in the whole affine ambient space" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "3819cb0c", + "metadata": {}, + "outputs": [], + "source": [ + "is_prime(ideal_sheaf(E1)) # in this case the exceptional divisor is prime" + ] + }, + { + "cell_type": "markdown", + "id": "16140bbc", + "metadata": {}, + "source": [ + "We can perform (parts of) the usual operations for the blowup morphism `pr1`, such as e.g. pullback:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2756772b", + "metadata": {}, + "outputs": [], + "source": [ + "W_cov = codomain(pr1) # a `CoveredScheme` with only one chart `W`\n", + "II = ideal_sheaf(W_cov, W, ideal(OO(W), t+1)) # the `AbsIdealSheaf` on `W_cov` for the fiber over `t = -1" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ea66a2a6", + "metadata": {}, + "outputs": [], + "source": [ + "pb1_II = pullback(pr1, II) # the ideal sheaf for the fiber over `t = -1` in the blowup `W1`\n", + "pb1_II(U_simp)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "c431a7a6", + "metadata": {}, + "outputs": [], + "source": [ + "is_prime(pb1_II)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "75279b55", + "metadata": {}, + "outputs": [], + "source": [ + "F1 = weil_divisor(pb1_II) # the fiber as a `AbsWeilDivisor`\n", + "F1 = irreducible_decomposition(F1) # ...decomposed into irreducible components" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "002bb1da", + "metadata": {}, + "outputs": [], + "source": [ + "Q1, Q2 = components(F1);" + ] + }, + { + "cell_type": "markdown", + "id": "d72801d7", + "metadata": {}, + "source": [ + "Let us compare the components of the fiber with the exceptional divisor and the strict transform of the fiber." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2b03afa4", + "metadata": {}, + "outputs": [], + "source": [ + "str1_II = strict_transform(pr1, II);\n", + "is_prime(str1_II)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "27e6b1e3", + "metadata": {}, + "outputs": [], + "source": [ + "ideal_sheaf(E1) == Q1" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "933d164d", + "metadata": {}, + "outputs": [], + "source": [ + "ideal_sheaf(E1) == Q2" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d2fffd84", + "metadata": {}, + "outputs": [], + "source": [ + "str1_II == Q1" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "88698429", + "metadata": {}, + "outputs": [], + "source": [ + "str1_II == Q2" + ] + }, + { + "cell_type": "markdown", + "id": "c4057cd2", + "metadata": {}, + "source": [ + "For the fun of it, let us have a look at the `weil_divisor` associated to the square of the fiber ideal" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2883a200", + "metadata": {}, + "outputs": [], + "source": [ + "JJ = pb1_II^2\n", + "@time irreducible_decomposition(weil_divisor(JJ))" + ] + }, + { + "cell_type": "markdown", + "id": "f1d22164", + "metadata": {}, + "source": [ + "How do the components of the fiber over `t = -1` intersect?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "38dab652", + "metadata": {}, + "outputs": [], + "source": [ + "intersect(weil_divisor(Q1), weil_divisor(Q2)) # A custom method for Weil divisors on surfaces!" + ] + }, + { + "cell_type": "markdown", + "id": "98c9f73c", + "metadata": {}, + "source": [ + "How do we actually get this number?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "46d6bdbb", + "metadata": {}, + "outputs": [], + "source": [ + "JJ = Q1 + Q2; # the sum of two ideal sheaves; will be 0-dimensional in this case\n", + "Z = AlgebraicCycle(W1, ZZ, IdDict([JJ=>one(ZZ)])) # the associated `AbsAlgebraicCycle` with ZZ-coefficients" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e2bf0ef8", + "metadata": {}, + "outputs": [], + "source": [ + "irreducible_decomposition(Z) # decomposition into prime components with multiplicities" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "6d124b9d", + "metadata": {}, + "outputs": [], + "source": [ + "integral(ans) # pushforward to the point `spec(kk)`" + ] + }, + { + "cell_type": "markdown", + "id": "588793ae", + "metadata": {}, + "source": [ + "# References\n", + "\n", + " * **[Ful98]**: Fulton, W., Intersection Theory, Springer, 1998" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Julia 1.10.0", + "language": "julia", + "name": "julia-1.10" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.10.0" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} From 1f3e270a830b5142f06263899039e0528fc5ac1a Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Sun, 17 Nov 2024 19:16:07 +0100 Subject: [PATCH 05/41] Add data for an automorphism. --- experimental/Durham2024/material/coord_imgs.mrdi | 1 + 1 file changed, 1 insertion(+) create mode 100644 experimental/Durham2024/material/coord_imgs.mrdi diff --git a/experimental/Durham2024/material/coord_imgs.mrdi b/experimental/Durham2024/material/coord_imgs.mrdi new file mode 100644 index 000000000000..a82f6f3fcd4f --- /dev/null +++ b/experimental/Durham2024/material/coord_imgs.mrdi @@ -0,0 +1 @@ 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\ No newline at end of file From 03575a34903e7bb3aea0f04788fe46fb6ca037a8 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Sun, 17 Nov 2024 19:51:51 +0100 Subject: [PATCH 06/41] Add second part of the presentation. --- .../Durham2024/material/Durham_2024_II.ipynb | 818 ++++++++++++++++++ 1 file changed, 818 insertions(+) create mode 100644 experimental/Durham2024/material/Durham_2024_II.ipynb diff --git a/experimental/Durham2024/material/Durham_2024_II.ipynb b/experimental/Durham2024/material/Durham_2024_II.ipynb new file mode 100644 index 000000000000..6ce7ab6a2cd5 --- /dev/null +++ b/experimental/Durham2024/material/Durham_2024_II.ipynb @@ -0,0 +1,818 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "ed8afc8f", + "metadata": {}, + "source": [ + "# Algebraic Geometry in OSCAR: part II\n", + "\n", + "## A look into the OSCAR kitchen: What's cooking?\n", + "\n", + "Together with [Simon Brandhorst](https://www.math.uni-sb.de/ag/brandhorst/) we are currently working on finding an [automorphism of minimal entropy for Enriques surfaces](https://arxiv.org/pdf/1807.09452v2.pdf). To this end, we have implemented and automatized wide parts of the functionality for *elliptic surfaces*." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "c676d78b", + "metadata": {}, + "outputs": [], + "source": [ + "using Revise, Oscar" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "81757145", + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "kk = GF(113) # the finite field (Galois field) with 113 elements\n", + "kkt_poly, t_poly = kk[:t] # the polynomial ring in one variable `t` over `kk`\n", + "kkt_field = fraction_field(kkt_poly) # the fraction field of that polynomial ring\n", + "P, (x, y) = kkt_field[:x, :y]; # the bivariate polynomial ring over `kkt_field`\n", + "f = y^2 - x^3 - (1 - t_poly^8)*x # a Weierstrass equation for an elliptic curve\n", + "E = elliptic_curve(f, x, y) # the associated elliptic curve as an instance of `EllipticCurve`" + ] + }, + { + "cell_type": "markdown", + "id": "c8cb97c0", + "metadata": {}, + "source": [ + "We can construct an elliptic surface directly from `E`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "724727a5", + "metadata": {}, + "outputs": [], + "source": [ + "X = elliptic_surface(E, 2; resolution_strategy=:simultaneous)" + ] + }, + { + "cell_type": "markdown", + "id": "d2ff33ec", + "metadata": {}, + "source": [ + "Rational points on `E` will give rise to elements in the *Mordell-Weil lattice* of `X`. We have picked some points and make them known on `X`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "6b0cef5f", + "metadata": {}, + "outputs": [], + "source": [ + "t = kkt_field(t_poly);\n", + "\n", + "basis_mwl = [[62*t^4 + 6*t^3 + 106*t^2 + 75*t + 26, 66*t^6 + 67*t^5 + 37*t^4 + 60*t^3 + 67*t^2 + 101*t + 58], [112*t^4 + 97*t^2 + 98, 104*t^5 + 82*t^3 + 91*t], [t^4 + 112, 87*t^4 + 26], [8*t^4 + 17*t^3 + 11*t^2 + 70*t + 7, 39*t^6 + 17*t^5 + 60*t^4 + 64*t^3 + 41*t^2 + 29*t + 89], [50*t^4 + 104*t^2 + 72, 56*t^6 + 30*t^4 + 36*t^2 + 104], [31*t^4 + 45*t^3 + 60*t^2 + 59*t + 13, 43*t^6 + 18*t^5 + 67*t^4 + 75*t^3 + 10*t^2 + 44*t + 17], [105*t^4 + 43*t^3 + 61*t^2 + 17*t + 7, 20*t^6 + 33*t^5 + 53*t^4 + 9*t^3 + 63*t^2 + 70*t + 89], [62*t^4 + 75*t^3 + 8*t^2 + 107*t + 87, 66*t^6 + 103*t^5 + 103*t^4 + 63*t^3 + 46*t^2 + 37*t + 34]]\n", + "\n", + "basis_mwl = [E(i) for i in basis_mwl]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0acb4fca", + "metadata": {}, + "outputs": [], + "source": [ + "Oscar.set_mordell_weil_basis!(X, basis_mwl)" + ] + }, + { + "cell_type": "markdown", + "id": "f3a68ae5", + "metadata": {}, + "source": [ + "In order to see more of what's going on, we may set up the verbosity." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9a1030ef", + "metadata": {}, + "outputs": [], + "source": [ + "set_verbose_level(:EllipticSurface, 5)\n", + "set_verbose_level(:MorphismFromRationalFunctions, 5)" + ] + }, + { + "cell_type": "markdown", + "id": "22e56858", + "metadata": {}, + "source": [ + "For elliptic surfaces we can compute a basis for the **Neron-Severi group** whenever we have found a Mordell-Weil basis." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d9f24377", + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "# The following one command is only necessary for the correct bookkeeping with further code below; ignore this!\n", + "Oscar._trivial_lattice(X;reducible_singular_fibers_in_PP1=Oscar._reducible_fibers_disc(X;sort=false))\n", + "\n", + "# compute the full Neron-Severi lattice of `X`\n", + "alg_lat, _, NS = algebraic_lattice(X)\n", + "V = ambient_space(NS)" + ] + }, + { + "cell_type": "markdown", + "id": "2847fa4f", + "metadata": {}, + "source": [ + "The elements in `alg_lat` are divisors spanning the divisor class group up to *numerical equivalence*. \n", + "The big matrix is the *gram matrix* for their pairwise intersections." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2e0eba30", + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "alg_lat" + ] + }, + { + "cell_type": "markdown", + "id": "80ac901a", + "metadata": {}, + "source": [ + "Since `X` is a smooth, compact surface, the intersection of either two Weil divisors is well defined and we can compute pairwise intersection numbers." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "26cf3206", + "metadata": {}, + "outputs": [], + "source": [ + "intersect(alg_lat[1], alg_lat[12])" + ] + }, + { + "cell_type": "markdown", + "id": "35b118d7", + "metadata": {}, + "source": [ + "Even the self-intersection of all generators is known here! But unfortunately this is only known due to some tricks on elliptic surfaces..." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "070b85af", + "metadata": {}, + "outputs": [], + "source": [ + "intersect(alg_lat[1], alg_lat[1]) # the fiber of the elliptic fibration has self-intersection 0" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2f2aca54", + "metadata": {}, + "outputs": [], + "source": [ + "intersect(alg_lat[3], alg_lat[3]) # any rational curve on a K3-surface has self intersection -2" + ] + }, + { + "cell_type": "markdown", + "id": "06c22c84", + "metadata": {}, + "source": [ + "### A dirty speedup for the self-intersection of curves\n", + "\n", + "> **Task 1:** Extend the computation of self-intersection of curves on a smooth surface by means of the [adjunction formula](https://en.wikipedia.org/wiki/Adjunction_formula) and the [Riemann-Roch theorem](https://en.wikipedia.org/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem) for curves.\n", + "\n", + "Roughly this can be done as follows.\n", + "\n", + "> **Theorem (Riemann-Roch for curves):** Let $C$ be a smooth projective curve and $D$ a divisor on it. Then \n", + "$$\\chi(D) = h^0(\\mathcal O(D)) - h^1(\\mathcal O(D)) = \\mathrm{deg}(D) + 1 - g(C)$$\n", + "where $g(C) = h^0(\\omega_C)$ is the *genus* and $\\mathrm{deg}(D) = \\int_C c_1(\\mathcal O(D))$ is the degree of the divisor.\n", + "\n", + "Applying this to the canonical divisor $D = \\omega_C$, we get the \n", + "well known formula $\\deg(\\omega_C) = 2g(C) - 2$. For the canonical bundle of $C$ we have a formula relating it to the canonical bundle of the ambient surface:\n", + "\n", + "> **Theorem (Adjunction formula):** Let $X$ be a smooth, projective surface and $i : C \\hookrightarrow X$ a smooth divisor on it. Then \n", + "$$ \\omega_C = i^* (\\omega_X \\otimes \\mathcal O(C)).$$\n", + "\n", + "Intersection theory tells us that $\\mathrm{deg}(i^*\\mathcal O(C)) = C.C$ is equal to the self-intersection of $C$ in $X$ and that the degree is logarithmic for line bundles on curves. Therefore \n", + "$$2g(C) - 2 = \\mathrm{deg}(\\omega_C) = \\mathrm{deg}(i^* \\omega_X) + \\deg(i^*\\mathcal O(C))) = K_X.C + C.C$$\n", + "where $K_X$ is the *canonical divisor* of the surface $X$.\n", + "\n", + "The algorithm would now proceed as follows:\n", + " * Is a canonical divisor of the surface known? If not, throw an error `can not compute self-intersection`.\n", + " * Is the curve smooth and in general position w.r.t $K_X$? If not, throw an error.\n", + " * Compute the genus of $C$.\n", + " * Return $2g(C) - 2 - C.K_X$.\n", + " \n", + "For the computation of the genus we may fall back to [implementations in Singular](https://www.singular.uni-kl.de/Manual/4-0-3/sing_1274.htm)." + ] + }, + { + "cell_type": "markdown", + "id": "25e400a2", + "metadata": {}, + "source": [ + "## Weil- and Cartier divisors\n", + "\n", + "> **Definition (algebraic cycle):** Let $X$ be a scheme. An *algebraic cycle* of dimension $k$ is a formal sum \n", + "$$ \\sum_{i} a_i \\cdot [V_i]$$\n", + "where $V_i \\subset X$ are $k$-dimensional *subvarieties* of $X$. \n", + "\n", + "Following **[Ful98]** we will only speak of a variety for reduced and irreducible schemes. The algebraic cycles of dimension $k$ form an Abelian group which will be denoted by $Z_k(X)$. \n", + "\n", + "> **Definition (Weil divisor):** Let $X$ be an equidimensional scheme of dimension $d$. A *Weil divisor* is an algebraic cycle on $X$ of dimension $d-1$.\n", + "\n", + "Some divisors are more special than others:\n", + "\n", + "> **Definition (Cartier divisor):** An *effective Cartier divisor* is given by an ideal sheaf $\\mathcal I$ on $X$ which is locally principal for some non-zero divisor. A *Cartier divisor* is a formal linear combination of effective Cartier divisors.\n", + "\n", + "In other words: For every point $p \\in X$ there exists an open neighborhood $U \\supset p$ and a non-zero divisor $f \\in \\mathcal O(U)$ such that $\\mathcal I(U) = \\langle f \\rangle \\subset \\mathcal O(U)$.\n", + "\n", + "For any two such ''local trivializations'' $(U, f)$ and $(V, g)$ of an effective Cartier divisor $C$ there is a unit $u \\in \\mathcal O(U \\cap V)$ such that $f|_{U \\cap V} = u \\cdot g|_{U \\cap V}$. These give rise to transition functions of a line bundle on $X$ which we will denote by $\\mathcal O(C)$. This extends naturally to all Cartier divisors by taking common refinements." + ] + }, + { + "cell_type": "markdown", + "id": "2d3548b6", + "metadata": {}, + "source": [ + "We also have a Cartier divisor floating around, coming as exceptional divisor of the resolution of singularities; but we first have to dig it out of the internals of the elliptic surface." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "43c96010", + "metadata": {}, + "outputs": [], + "source": [ + "Exc_amb = X.ambient_exceptionals[1] # a Cartier divisor on some smooth ambient space of `X`\n", + "Exc = pullback(X.inc_Y, Exc_amb) # the pullback of that divisor to `X` itself\n", + "first.(Exc.(affine_charts(X)))" + ] + }, + { + "cell_type": "markdown", + "id": "7a587975", + "metadata": {}, + "source": [ + "Any Cartier divisor gives rise to a Weil divisor:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "873b7c50", + "metadata": {}, + "outputs": [], + "source": [ + "Exc_weil = weil_divisor(Exc)\n", + "irreducible_decomposition(Exc_weil)" + ] + }, + { + "cell_type": "markdown", + "id": "7092773d", + "metadata": {}, + "source": [ + "The converse does not hold in general. However, it holds on *smooth* schemes and for every Weil divisor $D$ one may ask: ''Is $D$ actually Cartier?''\n", + "\n", + "> **Task 2:** Implement an algorithm which, given a Weil divisor $D$, decides whether $D$ is Cartier and, if so, computes a local trivialization for $D$.\n", + "\n", + "The point in this is, as we shall see later, that while the intersection of two Weil divisors is usually not well defined, the intersection of a Cartier- and a Weil divisor always is. In fact, Cartier divisors play a crucial role in the development of intersection theory." + ] + }, + { + "cell_type": "markdown", + "id": "c2237568", + "metadata": {}, + "source": [ + "### An automorphism of the K3 surface\n", + "\n", + "In our project with Simon Brandhorst we are constructing a very specific automorphism of a K3 surface (which is a lift to the universal cover of the Enriques surface automorphism proposed by Oguiso and Yu). K3 surfaces enjoy various favourable properties which allow us (more or less) to identify an automorphism $f \\colon X \\to X$ from its ''fingerprint'' given by the action on the Neron-Severi lattice\n", + "\n", + "$$ f_* \\colon \\mathrm{NS}(X) \\to \\mathrm{NS}(X).$$\n", + "\n", + "This should be thought of as something like a *hash function*: Any two given automorphisms $f$ and $g$ are very likely to agree if their actions on the Neron-Severi lattice do. However, given $f_*$ as above, it is almost impossible to reconstruct $X$ and $f$. This is basically what the whole paper with Simon is about!\n", + "\n", + "Let us have a look at the reconstruction of $f$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ab079123", + "metadata": {}, + "outputs": [], + "source": [ + "# the matrix representing the Oguiso-Yu automorphism w.r.t the basis of V\n", + "fOYonAlgX = QQ[5 1 0 -2 1 -1 0 -1 1 0 -1 0 -1 1 0 0 0 2; 2 -1 1 -2 2 0 1 -2 2 0 -2 0 -2 3 0 1 1 3; 4 1 0 -1 1 -1 0 -1 1 0 -1 0 -1 1 0 0 0 2; 5 1 0 -2 1 -1 0 -2 1 0 -1 0 -1 1 0 0 0 2; 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0; 5 1 0 -2 1 -1 0 -1 1 -1 -1 0 -1 1 0 0 0 2; 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 1 0 -2 1 0 0 -1 1 0 -1 0 -1 1 0 0 0 2; 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0; 10 -1 2 -7 5 -2 2 -4 5 1 -5 0 -5 7 0 2 2 9; 10 3 -1//2 -7//2 1//2 -3//2 -1//2 -5//2 1//2 -1//2 -2 0 -2 2 0 0 0 2; 7 1 1 -3 1 -1 0 -2 2 0 -3 0 -2 2 0 1 1 3; 4 0 1 -3 2 0 1 -3 1 1 -2 0 -3 3 1 1 0 3; 8 2 1 -3 1 -1 0 -2 2 0 -3 0 -2 2 -1 1 1 3; 10 0 2 -6 4 -2 1 -3 4 0 -5 0 -3 5 -1 2 2 7; 9 1 1//2 -9//2 5//2 -3//2 3//2 -7//2 5//2 1//2 -3 -1 -4 5 0 1 1 5; -3 -2 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1 1]" + ] + }, + { + "cell_type": "markdown", + "id": "0ba8189b", + "metadata": {}, + "source": [ + "So that is the fingerprint we need to realize. We have constructed a candidate for the automorphism; let's load it." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ad29e9ac", + "metadata": {}, + "outputs": [], + "source": [ + "coord_imgs = load(\"coord_imgs.mrdi\")\n", + "coord_imgs = [evaluate(fr, gens(ambient_coordinate_ring(weierstrass_chart_on_minimal_model(X)))) for fr in coord_imgs]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "693ffd41", + "metadata": {}, + "outputs": [], + "source": [ + "aut = morphism_from_rational_functions(X, X,\n", + " weierstrass_chart_on_minimal_model(X),\n", + " weierstrass_chart_on_minimal_model(X),\n", + " coord_imgs; check=false\n", + " );" + ] + }, + { + "cell_type": "markdown", + "id": "9e0b44e8", + "metadata": {}, + "source": [ + "Weil divisors are covariant in nature with respect to proper maps. \n", + "\n", + "> **Definition (proper pushforward):** Let $f \\colon X \\to Y$ be a proper morphism of algebraic schemes, $P \\in X$ a point of dimension $p$ and $Q = f(P) \\in Y$ its image. The pushforward of the cycle $[P] \\in Z_p(X)$ is defined as \n", + "$$ f_*([P]) = \\begin{cases} [R(P):R(Q)] \\cdot [Q] &\\mathrm{ if } \\quad \\dim Q = \\dim P \\\\ 0 & \\mathrm{ otherwise.} \\end{cases}$$\n", + "The general pushforward for algebraic cycles is defined by linear extension to $Z_p(X)$.\n", + "\n", + "Here we denote by $R(P)$ the field of rational functions on the subvariety of $X$ given by $P$ and vice versa for $Q$. \n", + "\n", + "For an *isomorphism* we can skip the computation of the degree of field extensions and just compute the image of a point." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "430bc097", + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "set_attribute!(aut, :is_isomorphism=>true)\n", + "D = alg_lat[3]\n", + "pf_D = pushforward(aut, D)" + ] + }, + { + "cell_type": "markdown", + "id": "0b6b4f7a", + "metadata": {}, + "source": [ + "> **Task 3:** Extend the functionality for pushforward of algebraic cycles to more than just isomorphisms.\n", + "\n", + "One possibility would, for instance, be blowdown maps where we know that they are proper and birational. More generally, we do not have a good method to compute the degree of the field extension directly. But one could certainly think of one or two dirty workarounds for this..." + ] + }, + { + "cell_type": "markdown", + "id": "af7ce9c2", + "metadata": {}, + "source": [ + "For K3 surfaces we can do more than just push forward a divisor: We can represent the image divisor in our chosen basis for the Neron-Severi lattice!" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ab332441", + "metadata": {}, + "outputs": [], + "source": [ + "rep_pf_D = basis_representation(X, pf_D)" + ] + }, + { + "cell_type": "markdown", + "id": "4d33ea45", + "metadata": {}, + "source": [ + "For elliptically fibered K3 surfaces we have an automated and optimized method to do everything at once." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "4e7f951d", + "metadata": {}, + "outputs": [], + "source": [ + "pf_aut = Oscar.pushforward_on_algebraic_lattices(aut) # CAUTION: This command takes some time for computation!" + ] + }, + { + "cell_type": "markdown", + "id": "116c7f3a", + "metadata": {}, + "source": [ + "That finding a basis representation for a divisor actually works is due to a rather remarkable property of K3 surfaces: The notions of rational-, algebraic-, and numerical equivalence agree. Consequently, we can compute a basis representation for any divisor from its intersection numbers with the generators of the Neron-Severi lattice.\n", + "\n", + "Note, however, that for a general surface it is almost impossible to determine whether or not two given divisors are even rationally equivalent!\n", + "\n", + "Before we move on, let us once compare the induced map on $\\mathrm{NS}(X)$ for the given automorphism `aut` with the fingerprint of `f`:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e0a1e098", + "metadata": {}, + "outputs": [], + "source": [ + "matrix(pf_aut)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "386e439a", + "metadata": {}, + "outputs": [], + "source": [ + "fOYonAlgX" + ] + }, + { + "cell_type": "markdown", + "id": "cb33f6ad", + "metadata": {}, + "source": [ + "As you can see, we did not get completely lucky this time. Some more adjustments seem to be needed." + ] + }, + { + "cell_type": "markdown", + "id": "b5215c37", + "metadata": {}, + "source": [ + "### A brief review of rational equivalence\n", + "\n", + "Let $X$ be a scheme, $U \\subset X$ an affine open subset, and $f \\in \\mathcal O(U)$ a function. \n", + "For a prime ideal $P \\subset \\mathcal O(U)$ of codimension one we define the *order* of $f$ at \n", + "the point $P$ as \n", + "\n", + "$$ \\mathrm{ord}_P(f) = \\mathrm{length}(\\mathcal O_{X, P}/\\langle f\\rangle)$$\n", + "\n", + "where $\\mathcal O_{X, P} = \\mathcal O(U)_P$ is the localization at the complement of $P$.\n", + "This ring is local of dimension one, so that the quotient by $\\langle h\\rangle$ is zero-dimensional and hence has finite length.\n", + "\n", + "\n", + "If $X$ is a variety, then this extends naturally to rational functions on $X$ and we get the following notion.\n", + "\n", + "> **Definition (principal divisor):** Let $W$ be a variety and $f \\in R(W)$ a rational function. The *principal divisor* associated to $f$ is \n", + "$$\\mathrm{div}(f) = \\sum_{P \\subset W} \\mathrm{ord}_P(f) \\cdot [P].$$\n", + "\n", + "Moreover, two algebraic cycles can be related by the following equivalence.\n", + "\n", + "> **Definition (rational equivalence):** Two algebraic cycles $D, E \\in Z_k(X)$ on a scheme $X$ are called *rationally equivalent* $D \\sim E$ if there exist a finite number of subvarieties $W_i \\subset X$ of dimension $k+1$ and rational functions $f_i \\in R(W_i)$ such that \n", + "$$ D - E = \\sum_i \\mathrm{div}(f_i).$$\n", + "\n", + "The quotient $Z_k(X)/\\sim$ will be denoted by $A_k(X)$, the $k$-th *Chow group* of $X$.\n", + "\n", + "> **Proposition (pushforward on Chow groups):** The pushforward for proper maps descends to a morphism of Chow groups.\n", + "\n", + "See e.g. **[Ful98]**, Theorem 1.4." + ] + }, + { + "cell_type": "markdown", + "id": "5072cc58", + "metadata": {}, + "source": [ + "We do already have a data type for rational functions on varieties:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "72fd6873", + "metadata": {}, + "outputs": [], + "source": [ + "RX = function_field(X)" + ] + }, + { + "cell_type": "markdown", + "id": "7c021124", + "metadata": {}, + "source": [ + "Elements of this field are represented on the different charts `U` of `X` as fractions of polynomials for the `ambient_coordinate_ring` of `U`. This is mostly due to the fact that we have no implementation of fraction fields for quotients of polynomial rings at this time. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "3aee41f5", + "metadata": {}, + "outputs": [], + "source": [ + "W = weierstrass_chart_on_minimal_model(X)\n", + "R = ambient_coordinate_ring(W)\n", + "(x, y, t) = gens(R)\n", + "f = RX(x) # x as a rational function on `X`; potentially with poles outside `W`" + ] + }, + { + "cell_type": "markdown", + "id": "24c660cc", + "metadata": {}, + "source": [ + "Let us look at a representative of $f$ in another chart:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e067eaf5", + "metadata": {}, + "outputs": [], + "source": [ + "simp_cov = Oscar.simplified_covering(X)\n", + "U = simp_cov[5]\n", + "f[U] # Note that we are using rectangular brackets for this at the moment!" + ] + }, + { + "cell_type": "markdown", + "id": "82e96ac0", + "metadata": {}, + "source": [ + "Note that representatives are not necessarily automatically reduced." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "be1f453d", + "metadata": {}, + "outputs": [], + "source": [ + "g = RX(first(gens(modulus(OO(W)))))\n", + "g[U]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "7586ae90", + "metadata": {}, + "outputs": [], + "source": [ + "is_zero(g)" + ] + }, + { + "cell_type": "markdown", + "id": "4e862dcf", + "metadata": {}, + "source": [ + "Determining the principal divisor associated to a rational function is something which is actually still missing from our implementations!\n", + "\n", + "> **Task 4:** Implement an algorithm which computes $\\mathrm{div}(f)$ for a rational function $f \\in R(X)$ on a variety $X$." + ] + }, + { + "cell_type": "markdown", + "id": "cf973d9f", + "metadata": {}, + "source": [ + "However, we already have a method to determine the order of vanishing of a rational function on a divisor which might be helpful." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0ed0ca2e", + "metadata": {}, + "outputs": [], + "source": [ + "I = ideal(OO(W), y)\n", + "decomp = primary_decomposition(I)\n", + "P = decomp[1][1]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "1e8d1cf7", + "metadata": {}, + "outputs": [], + "source": [ + "D = weil_divisor(ideal_sheaf(X, W, P))\n", + "order_of_vanishing(f, D)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e5deb4e7", + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "order_of_vanishing(f^(-2), D)" + ] + }, + { + "cell_type": "markdown", + "id": "61e1bdc5", + "metadata": {}, + "source": [ + "## More on Cartier divisors\n", + "\n", + "While Weil divisors are of homological nature (i.e. they have a covariant behaviour), Cartier divisors are cohomological: Given a morphism of schemes $f \\colon X \\to Y$ and a Cartier divisor $C$ on $Y$ there should be a natural notion of ''pulling back'' $C$. Unfortunately, this is not quite the case.\n", + "\n", + "### Cartier divisors on varieties\n", + "On a *variety* $X$ we can define Cartier divisors to be given by a collection $(U_i, f_i)$ where $f_i \\in R(X)$ is a rational function and for any pairs $U_i \\cap U_j$ the quotient $f_i/f_j \\in \\mathcal O(U_i \\cap U_j)^*$.\n", + "In theory, this spares us the detour to complete the monoid of effective Cartier divisors.\n", + "\n", + "Recall that to each Cartier divisor $C$ we can associate a line bundle $\\mathcal L = \\mathcal O(C)$. \n", + "A rational function $f \\in R(X)$ determines a *principal* Cartier divisor and we may call two Cartier divisors \n", + "*rationally equivalent*, if their difference is principal. \n", + "\n", + "> **Lemma:** On a variety $X$ there is a $1:1$-correspondence between Cartier divisors up to rational equivalence and isomorphism classes of line bundles. \n", + "\n", + "In order to get back a Cartier divisor from a line bundle $\\mathcal L$ we may construct a *rational section* $s$ of $\\mathcal L$: Start on one chart $U_0$ of a local trivialization of $\\mathcal L$ and extend the local algebraic section $e$ for this trivialization as a rational section to all of $X$. In the notation above, the associated Cartier divisor will then be given by the collection $(U_j, f_j/f_0)$. We denote it by $c_1(\\mathcal L)$. \n", + "\n", + "### Intersection of a Cartier divisor with an algebraic cycle\n", + "\n", + "Let $X$ be a scheme and $D = \\sum_i a_i \\cdot W_i \\in Z_k(X)$ an algebraic cycle with $W_i$ irreducible. For a Cartier divisor $C = \\{(U_j, f_j)\\}_{j=1}^N$ we may define the intersection $C \\cap D$ as follows.\n", + "\n", + " * We reduce to the case of irreducible cycles by $C \\cap (\\sum_i a_i \\cdot W_i) = \\sum_i a_i \\cdot (C \\cap W_i)$.\n", + " * For an irreducible cycle $\\iota \\colon W \\hookrightarrow X$ we set \n", + " $$C \\cap W := \\iota_*(c_1(\\iota^* \\mathcal O(C))\\cap [W]).$$\n", + " * $\\iota^*\\mathcal O(C)$ is the pullback of the line bundle $\\mathcal O(C)$ associated to $C$. \n", + " * $\\iota_*$ denotes the proper pushforward along the closed embedding $\\iota$.\n", + " * $[W]$ denotes the fundamental class of $[W]$. \n", + " * $c_1(\\mathcal L) \\cap [W]$ is the Weil divisor on $W$ associated to the Cartier divisor $c_1(\\mathcal L)$." + ] + }, + { + "cell_type": "markdown", + "id": "1d981f7f", + "metadata": {}, + "source": [ + "### More general intersections of divisors on surfaces\n", + "\n", + "The current implementation has a method `intersect(D::AbsAlgebraicCycle, C::CartierDivisor)`, but it is not fully implemented yet! Let's have a look at some examples." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f06621ab", + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "Z = intersect(alg_lat[1], Exc)\n", + "integral(Z)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "3867e729", + "metadata": {}, + "outputs": [], + "source": [ + "Z = intersect(alg_lat[2], Exc)\n", + "integral(Z)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "66b77bae", + "metadata": {}, + "outputs": [], + "source": [ + "Z = intersect(alg_lat[3], Exc)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "01dada74", + "metadata": {}, + "outputs": [], + "source": [ + "Z = intersect(alg_lat[12], Exc)\n", + "integral(Z)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "b10822bb", + "metadata": {}, + "outputs": [], + "source": [ + "Z_decomp = irreducible_decomposition(Z)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f960a263", + "metadata": {}, + "outputs": [], + "source": [ + "integral(ans)" + ] + }, + { + "cell_type": "markdown", + "id": "f0dcbb90", + "metadata": {}, + "source": [ + "This leads us to some more tricky tasks to achieve this week.\n", + "\n", + "> **Task 5:** Fill the gaps in the implementation of `intersect` for an `EffectiveCartierDivisor` and an `AbsAlgebraicCycle`.\n", + "\n", + "**Warning:** This is a deep dive! We will need some preparation for that.\n", + "\n", + "> **Task 6:** Speed up the implementation for `irreducible_decomposition`. \n", + "\n", + "It seems that a lot of time is spent in the generic method. But for many useful applications it should be possible to use more efficient shortcuts!\n", + "\n", + "> **Task 7:** Implement a method to construct the line bundle $\\mathcal O(C)$ associated to a Cartier divisor.\n", + "\n", + "This is rather technical. We do have support for coherent sheaves, though, but maybe for the purpose of intersection theory, we can avoid actually carrying out this nasty step?\n", + "\n", + "> **Task 8:** Implement the reconstruction of a Cartier divisor for a given line bundle $\\mathcal L$ on a variety $X$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "4893a0d8", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Julia 1.10.0", + "language": "julia", + "name": "julia-1.10" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.10.0" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} From 17cc1fb7112b69378e1a2946d110f04048df3692 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Wed, 20 Nov 2024 11:41:25 +0100 Subject: [PATCH 07/41] Allow return of components of algebraic cycles as cycles. --- experimental/Durham2024/src/Durham2024.jl | 2 ++ experimental/Durham2024/src/fixes.jl | 18 ++++++++++++++++++ experimental/Durham2024/test/fixes.jl | 11 +++++++++++ experimental/Durham2024/test/runtests.jl | 6 ++---- 4 files changed, 33 insertions(+), 4 deletions(-) create mode 100644 experimental/Durham2024/src/fixes.jl create mode 100644 experimental/Durham2024/test/fixes.jl diff --git a/experimental/Durham2024/src/Durham2024.jl b/experimental/Durham2024/src/Durham2024.jl index f71df7f5e83e..49a287dba54a 100644 --- a/experimental/Durham2024/src/Durham2024.jl +++ b/experimental/Durham2024/src/Durham2024.jl @@ -10,3 +10,5 @@ function pushforward(p::BlowupMorphism, W::AbsAlgebraicCycle) error("not implemented") end +include("fixes.jl") + diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl new file mode 100644 index 000000000000..a5722b56edd1 --- /dev/null +++ b/experimental/Durham2024/src/fixes.jl @@ -0,0 +1,18 @@ +function components(::Type{T}, D::AbsAlgebraicCycle) where {T <: AbsAlgebraicCycle} + X = scheme(D) + R = coefficient_ring(D) + return [AlgebraicCycle(X, R, IdDict{AbsIdealSheaf, elem_type(R)}([I=>one(R)]); check=false) for I in components(D)] +end + +function components(::Type{T}, D::AbsWeilDivisor) where {T <: AbsWeilDivisor} + X = scheme(D) + R = coefficient_ring(D) + return [WeilDivisor(X, R, IdDict{AbsIdealSheaf, elem_type(R)}([I=>one(R)]); check=false) for I in components(D)] +end + +function getindex(D::AbsAlgebraicCycle, C::AbsAlgebraicCycle) + comps = components(C) + @assert isone(length(comps)) + return D[first(comps)] +end + diff --git a/experimental/Durham2024/test/fixes.jl b/experimental/Durham2024/test/fixes.jl new file mode 100644 index 000000000000..59377edff845 --- /dev/null +++ b/experimental/Durham2024/test/fixes.jl @@ -0,0 +1,11 @@ +@testset "components of cycles" begin + IA2 = affine_space(QQ, [:x, :y]) + bl0 = blow_up(IA2, ideal(OO(IA2), gens(OO(IA2)))) + E = exceptional_divisor(bl0) + EW = weil_divisor(E) + c = first(components(WeilDivisor, EW)) + D = Oscar.underlying_cycle(EW) + cc = first(components(AlgebraicCycle, D)) + @test EW[c] == D[cc] +end + diff --git a/experimental/Durham2024/test/runtests.jl b/experimental/Durham2024/test/runtests.jl index 85a7ef3f1fa6..f643e81d832c 100644 --- a/experimental/Durham2024/test/runtests.jl +++ b/experimental/Durham2024/test/runtests.jl @@ -1,4 +1,2 @@ -@testset "Experimental template tests" begin - S = Oscar.ExampleStruct(5) - @test 5 == Oscar.my_access_func(S) -end +include("fixes.jl") + From 31e3b408a669242141199afed8cb2fc3875cc594 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Wed, 20 Nov 2024 17:26:49 +0100 Subject: [PATCH 08/41] Implement the Weil Divisor of a rational function. --- experimental/Durham2024/src/fixes.jl | 66 +++++++++++++++++++++++++++ experimental/Durham2024/test/fixes.jl | 8 ++++ 2 files changed, 74 insertions(+) diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl index a5722b56edd1..38b15267f551 100644 --- a/experimental/Durham2024/src/fixes.jl +++ b/experimental/Durham2024/src/fixes.jl @@ -16,3 +16,69 @@ function getindex(D::AbsAlgebraicCycle, C::AbsAlgebraicCycle) return D[first(comps)] end +scheme(f::VarietyFunctionFieldElem) = scheme(parent(f)) +variety(f::VarietyFunctionFieldElem) = variety(parent(f)) + +function _colength_in_localization(I::Ideal, P::Ideal) + return _colength_in_localization(saturated_ideal(I), saturated_ideal(P)) +end + +function _colength_in_localization(I::MPolyIdeal, P::MPolyIdeal) + R = base_ring(I) + @assert R === base_ring(P) + U = MPolyComplementOfPrimeIdeal(P) + L, loc = localization(R, U) + I_loc = loc(I) + F = free_module(L, 1) + IF, inc_IF = I_loc*F + M = cokernel(inc_IF) + return length(M) +end + + +function weil_divisor( + f::VarietyFunctionFieldElem; + ring::Ring=ZZ, covering::Covering=default_covering(scheme(f)) + ) + X = scheme(f) + ideal_dict = IdDict{AbsIdealSheaf, elem_type(ring)}() + for U in patches(covering) + inc_dict = IdDict{AbsIdealSheaf, elem_type(ring)}() + f_loc = f[U] + num = numerator(f_loc) + den = denominator(f_loc) + num_ideal = ideal(OO(U), num) + den_ideal = ideal(OO(U), den) + num_dec = primary_decomposition(num_ideal) + den_dec = primary_decomposition(den_ideal) + for (_, P) in num_dec + # If this component was already seen in another patch, skip it. + new_comp = PrimeIdealSheafFromChart(X, U, P) + any(new_comp == PP for PP in keys(ideal_dict)) && continue + c = _colength_in_localization(num_ideal, P) + inc_dict[new_comp] = c + end + for (_, P) in den_dec + # If this component was already seen in another patch, skip it. + any(P == PP for PP in keys(ideal_dict)) && continue + c = _colength_in_localization(num_ideal, P) + new_comp = PrimeIdealSheafFromChart(X, U, P) + if !any(new_comp == P for P in keys(inc_dict)) + is_zero(c) && continue + inc_dict[new_comp] = -c + else + d = inc_dict[new_comp] + if c == d + delete!(inc_dict, new_comp) + continue + end + inc_dict[new_comp] = d - c + end + end + for (pp, c) in inc_dict + ideal_dict[pp] = c + end + end + return WeilDivisor(X, ring, ideal_dict; check=false) +end + diff --git a/experimental/Durham2024/test/fixes.jl b/experimental/Durham2024/test/fixes.jl index 59377edff845..30e54c43a7d9 100644 --- a/experimental/Durham2024/test/fixes.jl +++ b/experimental/Durham2024/test/fixes.jl @@ -7,5 +7,13 @@ D = Oscar.underlying_cycle(EW) cc = first(components(AlgebraicCycle, D)) @test EW[c] == D[cc] + + X = domain(bl0) + FX = function_field(X) + U = first(affine_charts(X)) + x = first(gens(OO(U))) + f = FX(x) + D = weil_divisor(f) + @test 2*D == weil_divisor(f^2) end From be52898db55817a952727d50fbf6110a1f6c1310 Mon Sep 17 00:00:00 2001 From: StardustMath Date: Wed, 20 Nov 2024 16:53:53 +0000 Subject: [PATCH 09/41] Add functionality for canonical divisor --- experimental/Durham2024/src/fixes.jl | 9 ++++++ experimental/Durham2024/test/runtests2.jl | 34 +++++++++++++++++++++++ 2 files changed, 43 insertions(+) create mode 100644 experimental/Durham2024/test/runtests2.jl diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl index a5722b56edd1..ea33fade370a 100644 --- a/experimental/Durham2024/src/fixes.jl +++ b/experimental/Durham2024/src/fixes.jl @@ -16,3 +16,12 @@ function getindex(D::AbsAlgebraicCycle, C::AbsAlgebraicCycle) return D[first(comps)] end +function canonical_divisor(X:: AbsAffineScheme{<:Field, <:MPolyRing}) + return CartierDivisor(covered_scheme(X), ZZ) +end + + + + + + diff --git a/experimental/Durham2024/test/runtests2.jl b/experimental/Durham2024/test/runtests2.jl new file mode 100644 index 000000000000..5ad1a74b8f9a --- /dev/null +++ b/experimental/Durham2024/test/runtests2.jl @@ -0,0 +1,34 @@ +@testset "canonical divisor of a Kleinian singularity" begin +# We calculate the canonical divisor of the blow up of a Kleinian singularity in this document. +R, (x, y, z) = QQ[:x, :y, :z] +I = ideal(R, [x^3 - y*z]) + +IA3 = spec(R) +X, inc_X = sub(IA3, I) #X is the A2 Kleinian singularity +I_sing = ideal(R, gens(R)) + +pr1 = blow_up(IA3, I_sing) #blow up the affine space IA3 +bl_IA3 = domain(pr1) +E = exceptional_divisor(pr1) +bl_X, inc_bl_X, pr2 = strict_transform(pr1, inc_X) #bl_X is the blow up of X + +#the canonical divisor of bl_IA3 +bl_IA3_canonical_divisor = pullback(pr1, Oscar.canonical_divisor(IA3)) + (dim(IA3)-1)*E + +# the cartier divisor of IA3 that correspdonds to X +X_ideal = image_ideal(inc_X) +X_ideal_sheaf = ideal_sheaf(codomain(pr1), IA3, X_ideal) +X_divisor = EffectiveCartierDivisor(X_ideal_sheaf) + +#the cartier divisor of bl_IA3 that correspdonds to bl_X +bl_X_divisor = strict_transform(pr1, X_divisor) + +# the canonical divisor of bl_X +bl_X_canonical_divisor = pullback(inc_bl_X, bl_IA3_canonical_divisor + bl_X_divisor) +end + + + + + + From 1c23f54869b0a640254b8a878c5db62183df7338 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Thu, 21 Nov 2024 11:23:56 +0100 Subject: [PATCH 10/41] Introduce method for moving divisors (WIP). --- experimental/Durham2024/src/fixes.jl | 23 +++++++++++++++++++++++ 1 file changed, 23 insertions(+) diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl index 38b15267f551..7443ff0d551f 100644 --- a/experimental/Durham2024/src/fixes.jl +++ b/experimental/Durham2024/src/fixes.jl @@ -82,3 +82,26 @@ function weil_divisor( return WeilDivisor(X, ring, ideal_dict; check=false) end +function move_divisor(D::AbsWeilDivisor) + X = scheme(D) + is_zero(D) && return D + @assert is_prime(D) "divisor needs to be prime" + P = first(components(D)) + i = findfirst(U->!isone(P(U)), affine_charts(X)) + i === nothing && error("divisor is trivial") + U = affine_charts(X)[i] + I = P(U) + L, loc = localization(OO(U), complement_of_prime_ideal(I)) + LP = loc(I) + g = gens(saturated_ideal(I)) + g = sort!(g, by=total_degree) + i = findfirst(f->(ideal(L, f) == LP), g) + x = g[i] + FX = function_field(X) + f = FX(x) + return D - weil_divisor(f) +end + + + + From 2422acbdbb116614729a49b32f1bb72dabc29998 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Thu, 21 Nov 2024 11:32:54 +0100 Subject: [PATCH 11/41] Extend irreducible_decomposition to gather components. --- .../Schemes/Divisors/AlgebraicCycles.jl | 19 ++++++++++++++++++- 1 file changed, 18 insertions(+), 1 deletion(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl index 79660467c47f..01d3276b69e7 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl @@ -454,7 +454,24 @@ Return a cycle ``E`` equal to ``D`` but as a formal sum ``E = ∑ₖ aₖ ⋅ I where the `components` ``Iₖ`` of ``E`` are all sheaves of prime ideals. """ function irreducible_decomposition(D::AbsAlgebraicCycle) - all(is_prime, keys(coefficient_dict(D))) && return D + if all(is_prime, keys(coefficient_dict(D))) + coeff_dict = copy(coefficient_dict(D)) + for (P, c) in coeff_dict + for (Q, d) in coeff_dict + P === Q && continue + P != Q && continue + # P == Q from here onwards + if c == -d + delete!(coeff_dict, P) + delete!(coeff_dict, Q) + else + coefficient_dict(D)[P] = d + c + delete!(coeff_dict, Q) + end + end + end + return AlgebraicCycle(scheme(D), coefficient_ring(D), coeff_dict; check=false) + end result = zero(D) for (I, a) in coefficient_dict(D) next_dict = IdDict{AbsIdealSheaf, elem_type(coefficient_ring(D))}() From 14e610fef269c362ff3befbba3a438e50d881598 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Thu, 21 Nov 2024 11:46:28 +0100 Subject: [PATCH 12/41] Fix up pullback of Cartier divisors. --- experimental/Schemes/src/Auxiliary.jl | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/experimental/Schemes/src/Auxiliary.jl b/experimental/Schemes/src/Auxiliary.jl index b8cf7d827365..1e245b36460e 100644 --- a/experimental/Schemes/src/Auxiliary.jl +++ b/experimental/Schemes/src/Auxiliary.jl @@ -128,12 +128,11 @@ end function pullback(f::AbsCoveredSchemeMorphism, C::CartierDivisor) R = coefficient_ring(C) - C = CartierDivisor(domain(f), R) - pb = pullback(f) - for (c,D) in coefficient_dict(C) - C += c*pb(C) + result = CartierDivisor(domain(f), R) + for (D, c) in coefficient_dict(C) + result += c*pullback(f, D) end - return C + return result end function pullback(f::AbsCoveredSchemeMorphism, CC::Covering) From 31c45d4ebc34f5ec9e8179a31736c4ee8fb23928 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Thu, 21 Nov 2024 11:47:06 +0100 Subject: [PATCH 13/41] Disable tests for the moment. --- experimental/Durham2024/test/runtests2.jl | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/experimental/Durham2024/test/runtests2.jl b/experimental/Durham2024/test/runtests2.jl index 5ad1a74b8f9a..44b01e936243 100644 --- a/experimental/Durham2024/test/runtests2.jl +++ b/experimental/Durham2024/test/runtests2.jl @@ -24,7 +24,8 @@ X_divisor = EffectiveCartierDivisor(X_ideal_sheaf) bl_X_divisor = strict_transform(pr1, X_divisor) # the canonical divisor of bl_X -bl_X_canonical_divisor = pullback(inc_bl_X, bl_IA3_canonical_divisor + bl_X_divisor) +div_sum = bl_IA3_canonical_divisor + bl_X_divisor +#bl_X_canonical_divisor = pullback(inc_bl_X, div_sum) end From 02293002d2a6dac94a9dc7099b8003539b3fb4f1 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Thu, 21 Nov 2024 16:05:21 +0100 Subject: [PATCH 14/41] Fix up moving of divisors. --- experimental/Durham2024/src/fixes.jl | 20 +++++++++++--------- 1 file changed, 11 insertions(+), 9 deletions(-) diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl index 7443ff0d551f..784aefd37a46 100644 --- a/experimental/Durham2024/src/fixes.jl +++ b/experimental/Durham2024/src/fixes.jl @@ -60,19 +60,21 @@ function weil_divisor( end for (_, P) in den_dec # If this component was already seen in another patch, skip it. - any(P == PP for PP in keys(ideal_dict)) && continue - c = _colength_in_localization(num_ideal, P) new_comp = PrimeIdealSheafFromChart(X, U, P) - if !any(new_comp == P for P in keys(inc_dict)) + any(new_comp == PP for PP in keys(ideal_dict)) && continue + c = _colength_in_localization(num_ideal, P) + key_list = collect(keys(inc_dict)) + k = findfirst(==(new_comp), key_list) + if k === nothing is_zero(c) && continue inc_dict[new_comp] = -c else - d = inc_dict[new_comp] + d = inc_dict[key_list[k]] if c == d - delete!(inc_dict, new_comp) + delete!(inc_dict, key_list[k]) continue end - inc_dict[new_comp] = d - c + inc_dict[key_list[k]] = d - c end end for (pp, c) in inc_dict @@ -97,9 +99,9 @@ function move_divisor(D::AbsWeilDivisor) g = sort!(g, by=total_degree) i = findfirst(f->(ideal(L, f) == LP), g) x = g[i] - FX = function_field(X) - f = FX(x) - return D - weil_divisor(f) + kk = base_ring(U) + f = function_field(X)(x) + return irreducible_decomposition(D - weil_divisor(f)) end From f44252c5e123d565976cace3ab7d95f372732225 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Thu, 21 Nov 2024 16:05:37 +0100 Subject: [PATCH 15/41] Add tests. --- experimental/Durham2024/test/fixes.jl | 19 +++++++++++++++++++ 1 file changed, 19 insertions(+) diff --git a/experimental/Durham2024/test/fixes.jl b/experimental/Durham2024/test/fixes.jl index 30e54c43a7d9..bcfcf1d38433 100644 --- a/experimental/Durham2024/test/fixes.jl +++ b/experimental/Durham2024/test/fixes.jl @@ -15,5 +15,24 @@ f = FX(x) D = weil_divisor(f) @test 2*D == weil_divisor(f^2) + + (s, x, y) = ambient_coordinates(U) + I = ideal_sheaf(E) + IU = I(U) + @test OO(U)(x) in IU + @test OO(U)(y) in IU + @test !(OO(U)(s) in IU) + + f = FX(x) + W = weil_divisor(f) + I1, I2 = components(W) + @test I == I1 || I == I2 + @test I != I1 || I != I2 + + D = weil_divisor(E) + D2 = irreducible_decomposition(D - W) + @test integral(intersect(D2, E)) == -1 + + @test integral(intersect(Oscar.move_divisor(D), E)) == -1 end From ff6c7c24db2c54d2ca378b457b97cebf3fe68352 Mon Sep 17 00:00:00 2001 From: StardustMath Date: Thu, 21 Nov 2024 15:42:54 +0000 Subject: [PATCH 16/41] self intersection via adjunction --- experimental/Durham2024/src/fixes.jl | 4 +++- experimental/Durham2024/test/runtests.jl | 1 + experimental/Durham2024/test/runtests2.jl | 26 +++++++++++------------ 3 files changed, 16 insertions(+), 15 deletions(-) diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl index ea33fade370a..91185ac435e1 100644 --- a/experimental/Durham2024/src/fixes.jl +++ b/experimental/Durham2024/src/fixes.jl @@ -20,7 +20,9 @@ function canonical_divisor(X:: AbsAffineScheme{<:Field, <:MPolyRing}) return CartierDivisor(covered_scheme(X), ZZ) end - +function self_intersection_via_adjunction(K_X::AbsWeilDivisor, C::AbsWeilDivisor, g::Int) + return 2g-2-intersect(K_X, C) +end diff --git a/experimental/Durham2024/test/runtests.jl b/experimental/Durham2024/test/runtests.jl index f643e81d832c..7001b1965b7f 100644 --- a/experimental/Durham2024/test/runtests.jl +++ b/experimental/Durham2024/test/runtests.jl @@ -1,2 +1,3 @@ include("fixes.jl") +include("runtests2.jl") diff --git a/experimental/Durham2024/test/runtests2.jl b/experimental/Durham2024/test/runtests2.jl index 44b01e936243..1980ada89f85 100644 --- a/experimental/Durham2024/test/runtests2.jl +++ b/experimental/Durham2024/test/runtests2.jl @@ -12,20 +12,18 @@ bl_IA3 = domain(pr1) E = exceptional_divisor(pr1) bl_X, inc_bl_X, pr2 = strict_transform(pr1, inc_X) #bl_X is the blow up of X -#the canonical divisor of bl_IA3 -bl_IA3_canonical_divisor = pullback(pr1, Oscar.canonical_divisor(IA3)) + (dim(IA3)-1)*E - -# the cartier divisor of IA3 that correspdonds to X -X_ideal = image_ideal(inc_X) -X_ideal_sheaf = ideal_sheaf(codomain(pr1), IA3, X_ideal) -X_divisor = EffectiveCartierDivisor(X_ideal_sheaf) - -#the cartier divisor of bl_IA3 that correspdonds to bl_X -bl_X_divisor = strict_transform(pr1, X_divisor) - -# the canonical divisor of bl_X -div_sum = bl_IA3_canonical_divisor + bl_X_divisor -#bl_X_canonical_divisor = pullback(inc_bl_X, div_sum) +EX = pullback(inc_bl_X, E) +EX_weil = weil_divisor(EX) +EX_weil_decom = irreducible_decomposition(EX_weil) +C1 = components(WeilDivisor, EX_weil_decom)[1] +C2 = components(WeilDivisor, EX_weil_decom)[2] + +bl_X_canonical_divisor = weil_divisor(bl_X, ZZ) +Z11 = Oscar.self_intersection_via_adjunction(bl_X_canonical_divisor, C1, 0) +Z22 = Oscar.self_intersection_via_adjunction(bl_X_canonical_divisor, C2, 0) +Z12 = intersect(C1, C2) +Z21 = intersect(C2, C1) +Intersection_matrix = ZZ[Z11 Z12; Z21 Z22] end From ba63725603683868c2c75f7d07e371fb6c02346e Mon Sep 17 00:00:00 2001 From: Erroxe Date: Thu, 21 Nov 2024 16:15:20 +0000 Subject: [PATCH 17/41] added embedding function and example --- experimental/Durham2024/src/Durham2024.jl | 1 + .../Durham2024/src/erroxefunctions.jl | 8 ++ experimental/Durham2024/test/intersections.jl | 89 +++++++++++++++++++ 3 files changed, 98 insertions(+) create mode 100644 experimental/Durham2024/src/erroxefunctions.jl create mode 100644 experimental/Durham2024/test/intersections.jl diff --git a/experimental/Durham2024/src/Durham2024.jl b/experimental/Durham2024/src/Durham2024.jl index 49a287dba54a..00e6b16f5372 100644 --- a/experimental/Durham2024/src/Durham2024.jl +++ b/experimental/Durham2024/src/Durham2024.jl @@ -11,4 +11,5 @@ function pushforward(p::BlowupMorphism, W::AbsAlgebraicCycle) end include("fixes.jl") +include("erroxefunctions.jl") diff --git a/experimental/Durham2024/src/erroxefunctions.jl b/experimental/Durham2024/src/erroxefunctions.jl new file mode 100644 index 000000000000..c35f4c9ec980 --- /dev/null +++ b/experimental/Durham2024/src/erroxefunctions.jl @@ -0,0 +1,8 @@ +function embedding(C::EffectiveCartierDivisor) + I=ideal_sheaf(C); + C1,inc_C = sub(I); + return inc_C +end + +function reduce_to_surface() +end \ No newline at end of file diff --git a/experimental/Durham2024/test/intersections.jl b/experimental/Durham2024/test/intersections.jl new file mode 100644 index 000000000000..28d719fa917d --- /dev/null +++ b/experimental/Durham2024/test/intersections.jl @@ -0,0 +1,89 @@ +P3 = projective_space(QQ,[:x0,:x1,:x2,:x3]); +(x0,x1,x2,x3) = homogeneous_coordinates(P3); +R=homogeneous_coordinate_ring(P3); +######################################## +I = ideal(R,[x0,x3]); +#L,inc_L = sub(P3,I); +I1 = ideal(R,[x0-x1,x2]); +#L1,inc_L1 = sub(P3, I1); +I2 = ideal(R,[x1,x2]); +#L2,inc_L2 = sub(P3,ideal(R,[x1,x2])); +I3 = ideal(R,[x1,x2-x3]); +#L3,inc_L3 = sub(P3,ideal(R,[x1,x2-x3])); +# (X2,E2,inc_E2) = blowup(inc_L2); This does not work. +IH = ideal(x0-x2); +########################################## +Y = covered_scheme(P3); +II1=ideal_sheaf(P3,I1); +II2=ideal_sheaf(P3,I2); +II3=ideal_sheaf(P3,I3); +II=ideal_sheaf(P3,I); +IIH=ideal_sheaf(P3,IH); +H=effective_cartier_divisor(IIH); +############################################### +#We do the first blowup, by blowing up L +bl = blow_up(Y,II); +X= domain(bl); +II11=strict_transform(bl,II1); +II12=strict_transform(bl,II2); +II13=strict_transform(bl,II3); +H1=strict_transform(bl,H); +E=exceptional_divisor(bl); +################################################## +#We do the second blowup, by blowing up L2 +bl2 = blow_up(X,II12); +X2= domain(bl2); +II21=strict_transform(bl2,II11); +II23=strict_transform(bl2,II13); +H2=strict_transform(bl2,H1); +EE=strict_transform(bl2,E); +E2=exceptional_divisor(bl2); +################################################### +#We do the third blowup, by blowing up L1 +bl1 = blow_up(X2,II21); +X1= domain(bl1); +II33=strict_transform(bl1,II23); +H3=strict_transform(bl1,H2); +EEE=strict_transform(bl1,EE); +EE2=strict_transform(bl1,E2); +E1=exceptional_divisor(bl1); +#################################################### +#We do the fourth blowup, by blowing up L3 +bl3 = blow_up(X1,II33); +X_final= domain(bl3); +H_final = strict_transform(bl3,H3); +E_final=strict_transform(bl3,EEE); +E2_final=strict_transform(bl3,EE2); +E1_final=strict_transform(bl3,E1); +E3_final=exceptional_divisor(bl3); +################################################### +Int1=intersect(E_final,E1_final); +Int23=intersect(E2_final,E3_final); +Int21=intersect(E2_final,E1_final); +E2E3E1=intersect(Int23,E1_final); +integral(E2E3E1); +################################################## +#Now we want to do intersections inside the surface. Lets start with E2. +inc_E2 = embedding(E2_final); +l21=pullback(inc_E2,E1_final); +l23=pullback(inc_E2,E3_final); +Int123=intersect(l21,l23); +integral(Int123); +##################################################### +inc_E1 = embedding(E1_final); +l21=pullback(inc_E2,E1_final); +l23=pullback(inc_E2,E3_final); +Int123=intersect(l21,l23); +integral(Int123); +kX=-4*H_final+E1_final+E2_final+E3_final+E_final; +################################################### +inc_H = embedding(H_final); +l21=pullback(inc_E2,E1_final); +l23=pullback(inc_E2,E3_final); +Int123=intersect(l21,l23); +integral(Int123); + + + + + From b0a2c91fd66e3d4d6025a467b5eb95d2c9eee91d Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Thu, 21 Nov 2024 17:55:35 +0100 Subject: [PATCH 18/41] Fix up signature of self_intersection... --- experimental/Durham2024/src/fixes.jl | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl index 91185ac435e1..3bba8f9decad 100644 --- a/experimental/Durham2024/src/fixes.jl +++ b/experimental/Durham2024/src/fixes.jl @@ -20,8 +20,11 @@ function canonical_divisor(X:: AbsAffineScheme{<:Field, <:MPolyRing}) return CartierDivisor(covered_scheme(X), ZZ) end -function self_intersection_via_adjunction(K_X::AbsWeilDivisor, C::AbsWeilDivisor, g::Int) - return 2g-2-intersect(K_X, C) +function self_intersection_via_adjunction(K_X::CartierDivisor, C::EffectiveCartierDivisor, g::Int) + return 2g-2-intersect(K_X, C) +end +function self_intersection_via_adjunction(K_X::CartierDivisor, C::CartierDivisor, g::Int) + return 2g-2-sum(a*intersect(K_X, comp) for (a, comp) in coefficient_dict(C); init=0) end From fd8d9303e5d7d606cac18b50e9d49ed3cae83d00 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Fri, 22 Nov 2024 11:56:54 +0100 Subject: [PATCH 19/41] WIP on debugging. --- experimental/Durham2024/src/fixes.jl | 63 +++++++++++++++++++++++++--- 1 file changed, 58 insertions(+), 5 deletions(-) diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl index 8737b0846c0b..c1807af467a3 100644 --- a/experimental/Durham2024/src/fixes.jl +++ b/experimental/Durham2024/src/fixes.jl @@ -43,6 +43,7 @@ function _colength_in_localization(I::MPolyIdeal, P::MPolyIdeal) U = MPolyComplementOfPrimeIdeal(P) L, loc = localization(R, U) I_loc = loc(I) + @assert !isone(I_loc) F = free_module(L, 1) IF, inc_IF = I_loc*F M = cokernel(inc_IF) @@ -76,7 +77,7 @@ function weil_divisor( # If this component was already seen in another patch, skip it. new_comp = PrimeIdealSheafFromChart(X, U, P) any(new_comp == PP for PP in keys(ideal_dict)) && continue - c = _colength_in_localization(num_ideal, P) + c = _colength_in_localization(den_ideal, P) key_list = collect(keys(inc_dict)) k = findfirst(==(new_comp), key_list) if k === nothing @@ -98,10 +99,15 @@ function weil_divisor( return WeilDivisor(X, ring, ideal_dict; check=false) end -function move_divisor(D::AbsWeilDivisor) +function move_divisor(D::AbsWeilDivisor; check::Bool=false) X = scheme(D) is_zero(D) && return D - @assert is_prime(D) "divisor needs to be prime" + + if !is_prime(D) + R = coefficient_ring(D) + return sum(a*move_divisor(WeilDivisor(D, R; check=false)) for (D, a) in coefficient_dict(irreducible_decomposition(D)); init=WeilDivisor(X, R)) + end + P = first(components(D)) i = findfirst(U->!isone(P(U)), affine_charts(X)) i === nothing && error("divisor is trivial") @@ -113,9 +119,56 @@ function move_divisor(D::AbsWeilDivisor) g = sort!(g, by=total_degree) i = findfirst(f->(ideal(L, f) == LP), g) x = g[i] - kk = base_ring(U) - f = function_field(X)(x) + f = function_field(X; check)(x) return irreducible_decomposition(D - weil_divisor(f)) end +function is_zero(D::AbsAlgebraicCycle) + return all(is_zero(c) || is_one(I) for (I, c) in coefficient_dict(D)) +end + +function intersect(C::CartierDivisor, D::AbsWeilDivisor) + X = scheme(C) + @assert X === scheme(D) + R = coefficient_ring(C) + @assert R === coefficient_ring(D) + result = AlgebraicCycle(X, R) + for (E, c) in coefficient_dict(C) + result = result + c*intersect(E, D) + end + return result +end + +function intersect(E::EffectiveCartierDivisor, D::AbsWeilDivisor; check::Bool=true) + X = scheme(E) + @assert X === scheme(D) + R = coefficient_ring(D) + result = AlgebraicCycle(X, R) + DD = irreducible_decomposition(D) + for (P, a) in coefficient_dict(DD) + _, inc_P = sub(P) + if is_zero(pullback(inc_P, ideal_sheaf(E))) + P_moved = move_divisor(WeilDivisor(X, R, + IdDict{AbsIdealSheaf, elem_type(R)}( + [P=>one(R)]); check=false); check=false) + result = result + a*intersect(E, P_moved) + else + result = result + a*AlgebraicCycle(X, R, + IdDict{AbsIdealSheaf, elem_type(R)}( + [P + ideal_sheaf(E)=>one(R)]); check=false) + end + end + return result +end +function is_zero(II::AbsIdealSheaf) + return all(iszero(II(U)) for U in affine_charts(scheme(II))) +end + +function +(C::CartierDivisor, E::EffectiveCartierDivisor) + return C + 1*E +end + +function -(C::CartierDivisor, E::EffectiveCartierDivisor) + return C - 1*E +end From a9a14b01b9a6e92ffab255fe3cffdb0e745ce8a8 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Fri, 22 Nov 2024 17:56:00 +0100 Subject: [PATCH 20/41] WIP [no ci]. --- experimental/Durham2024/src/fixes.jl | 22 ++++++++++++++++++++++ 1 file changed, 22 insertions(+) diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl index c1807af467a3..53b1f44260e8 100644 --- a/experimental/Durham2024/src/fixes.jl +++ b/experimental/Durham2024/src/fixes.jl @@ -37,12 +37,34 @@ function _colength_in_localization(I::Ideal, P::Ideal) return _colength_in_localization(saturated_ideal(I), saturated_ideal(P)) end +function _find_principal_generator(I::MPolyLocalizedIdeal) + L = base_ring(I) + g = gens(I) + g = sort!(g, by=x->total_degree(lifted_numerator(x))) + for x in g + is_zero(x) && continue + ideal(L, x) == I && return x + end + error("no principal generator found") +end + function _colength_in_localization(I::MPolyIdeal, P::MPolyIdeal) R = base_ring(I) @assert R === base_ring(P) U = MPolyComplementOfPrimeIdeal(P) L, loc = localization(R, U) I_loc = loc(I) + P_loc = loc(P) + x = _find_principal_generator(P_loc) + y = one(L) + k = 0 + while true + y in I && return k + y = y*x + k += 1 + end + + @assert !isone(I_loc) F = free_module(L, 1) IF, inc_IF = I_loc*F From b16cf8f2eb98a56360fff5a56824295c252a6194 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 00:27:44 +0100 Subject: [PATCH 21/41] Fix up irreducible_decomposition. --- .../Schemes/Divisors/AlgebraicCycles.jl | 33 +++++++++++-------- 1 file changed, 20 insertions(+), 13 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl index 01d3276b69e7..74b4db60e52e 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl @@ -454,24 +454,31 @@ Return a cycle ``E`` equal to ``D`` but as a formal sum ``E = ∑ₖ aₖ ⋅ I where the `components` ``Iₖ`` of ``E`` are all sheaves of prime ideals. """ function irreducible_decomposition(D::AbsAlgebraicCycle) + @vprint :Divisors 4 "computing irreducible decomposition for $D" if all(is_prime, keys(coefficient_dict(D))) - coeff_dict = copy(coefficient_dict(D)) - for (P, c) in coeff_dict - for (Q, d) in coeff_dict - P === Q && continue - P != Q && continue - # P == Q from here onwards - if c == -d - delete!(coeff_dict, P) - delete!(coeff_dict, Q) - else - coefficient_dict(D)[P] = d + c - delete!(coeff_dict, Q) - end + @vprint :Divisors 4 " all components are already prime, cleaning up..." + buckets = Vector{Vector{AbsIdealSheaf}}() + for P in keys(coefficient_dict(D)) + found = false + for bucket in buckets + if P == first(bucket) + push!(bucket, P) + found = true + break + end end + !found && push!(buckets, [P]) + end + R = coefficient_ring(D) + coeff_dict = IdDict{AbsIdealSheaf, elem_type(R)}() + for bucket in buckets + c = sum(D[P] for P in bucket; init=zero(R)) + is_zero(c) && continue + coeff_dict[first(bucket)] = c end return AlgebraicCycle(scheme(D), coefficient_ring(D), coeff_dict; check=false) end + @vprint :Divisors 4 " decomposing components" result = zero(D) for (I, a) in coefficient_dict(D) next_dict = IdDict{AbsIdealSheaf, elem_type(coefficient_ring(D))}() From 3db78a0869454ac8a8b5c075fd4e3abe343592b0 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 00:32:31 +0100 Subject: [PATCH 22/41] Fix up moving of divisors, etc. --- experimental/Durham2024/src/fixes.jl | 59 +++++++++++++++------------- 1 file changed, 32 insertions(+), 27 deletions(-) diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl index 53b1f44260e8..75231033aceb 100644 --- a/experimental/Durham2024/src/fixes.jl +++ b/experimental/Durham2024/src/fixes.jl @@ -33,53 +33,52 @@ end scheme(f::VarietyFunctionFieldElem) = scheme(parent(f)) variety(f::VarietyFunctionFieldElem) = variety(parent(f)) +# Compute the colength of I_P in the localization R_P. +# This assumes that R itself is a domain and that I is of height 1. +# Then there exists a regular +# point on Spec(R) and hence R_P is also regular and a UFD. +# Being of height 1, I_P must then be principal. function _colength_in_localization(I::Ideal, P::Ideal) - return _colength_in_localization(saturated_ideal(I), saturated_ideal(P)) -end - -function _find_principal_generator(I::MPolyLocalizedIdeal) - L = base_ring(I) - g = gens(I) - g = sort!(g, by=x->total_degree(lifted_numerator(x))) - for x in g - is_zero(x) && continue - ideal(L, x) == I && return x - end - error("no principal generator found") -end - -function _colength_in_localization(I::MPolyIdeal, P::MPolyIdeal) R = base_ring(I) @assert R === base_ring(P) - U = MPolyComplementOfPrimeIdeal(P) + U = MPolyComplementOfPrimeIdeal(saturated_ideal(P)) L, loc = localization(R, U) I_loc = loc(I) + @assert base_ring(I_loc) === L P_loc = loc(P) x = _find_principal_generator(P_loc) y = one(L) k = 0 while true - y in I && return k + (y in I_loc) && return k y = y*x k += 1 end - - @assert !isone(I_loc) - F = free_module(L, 1) - IF, inc_IF = I_loc*F - M = cokernel(inc_IF) - return length(M) + return _colength_in_localization(saturated_ideal(I), saturated_ideal(P)) end +# same assumptions as above apply. +function _find_principal_generator(I::Union{<:MPolyLocalizedIdeal, <:MPolyQuoLocalizedIdeal}) + L = base_ring(I) + g = gens(I) + g = sort!(g, by=x->total_degree(lifted_numerator(x))) + for x in g + is_zero(x) && continue + ideal(L, x) == I && return x + end + error("no principal generator found") +end function weil_divisor( f::VarietyFunctionFieldElem; ring::Ring=ZZ, covering::Covering=default_covering(scheme(f)) ) + @vprint :Divisors 4 "calculating principal divisor for $f\n" X = scheme(f) ideal_dict = IdDict{AbsIdealSheaf, elem_type(ring)}() for U in patches(covering) + @vprint :Divisors 4 "doing patch $U\n" inc_dict = IdDict{AbsIdealSheaf, elem_type(ring)}() f_loc = f[U] num = numerator(f_loc) @@ -88,18 +87,25 @@ function weil_divisor( den_ideal = ideal(OO(U), den) num_dec = primary_decomposition(num_ideal) den_dec = primary_decomposition(den_ideal) + @vprint :Divisors 4 " numerator:\n" for (_, P) in num_dec + @vprint :Divisors 4 " $P\n" # If this component was already seen in another patch, skip it. new_comp = PrimeIdealSheafFromChart(X, U, P) + @vprint :Divisors 4 " $(any(new_comp == PP for PP in keys(ideal_dict)) ? "already found" : "new component")\n" any(new_comp == PP for PP in keys(ideal_dict)) && continue c = _colength_in_localization(num_ideal, P) + @vprint :Divisors 4 " multiplicity $c\n" inc_dict[new_comp] = c end + @vprint :Divisors 4 " denominator:\n" for (_, P) in den_dec # If this component was already seen in another patch, skip it. new_comp = PrimeIdealSheafFromChart(X, U, P) + @vprint :Divisors 4 " $(any(new_comp == PP for PP in keys(ideal_dict)) ? "already found" : "new component")\n" any(new_comp == PP for PP in keys(ideal_dict)) && continue c = _colength_in_localization(den_ideal, P) + @vprint :Divisors 4 " multiplicity $c\n" key_list = collect(keys(inc_dict)) k = findfirst(==(new_comp), key_list) if k === nothing @@ -166,7 +172,9 @@ function intersect(E::EffectiveCartierDivisor, D::AbsWeilDivisor; check::Bool=tr @assert X === scheme(D) R = coefficient_ring(D) result = AlgebraicCycle(X, R) + cpcd = copy(coefficient_dict(D)) DD = irreducible_decomposition(D) + cpcd = copy(coefficient_dict(DD)) for (P, a) in coefficient_dict(DD) _, inc_P = sub(P) if is_zero(pullback(inc_P, ideal_sheaf(E))) @@ -187,10 +195,7 @@ function is_zero(II::AbsIdealSheaf) return all(iszero(II(U)) for U in affine_charts(scheme(II))) end -function +(C::CartierDivisor, E::EffectiveCartierDivisor) - return C + 1*E -end - function -(C::CartierDivisor, E::EffectiveCartierDivisor) return C - 1*E end + From 281295783a81ee6dcc71134df47090b5917e22a8 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 00:32:49 +0100 Subject: [PATCH 23/41] Extend tests. --- experimental/Durham2024/test/intersections.jl | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/experimental/Durham2024/test/intersections.jl b/experimental/Durham2024/test/intersections.jl index 28d719fa917d..f370a48bd760 100644 --- a/experimental/Durham2024/test/intersections.jl +++ b/experimental/Durham2024/test/intersections.jl @@ -83,6 +83,13 @@ l23=pullback(inc_E2,E3_final); Int123=intersect(l21,l23); integral(Int123); +E2_sub, inc_E2 = sub(ideal_sheaf(E2_final)) +pb_E1 = pullback(inc_E2, E1_final) +pb_E1_weil = weil_divisor(pb_E1) +int1 = integral(intersect(pb_E1, pb_E1_weil)) +move = Oscar.move_divisor(pb_E1_weil) +int2 = integral(intersect(pb_E1, move)) +@test int1 == int2 From e2d8f281e3ee166c4ff4acf9f75c0499224ae146 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 13:03:04 +0100 Subject: [PATCH 24/41] Removing the course material. --- .../Durham2024/material/Durham_2024_I.ipynb | 641 -------------- .../Durham2024/material/Durham_2024_II.ipynb | 818 ------------------ .../Durham2024/material/coord_imgs.mrdi | 1 - 3 files changed, 1460 deletions(-) delete mode 100644 experimental/Durham2024/material/Durham_2024_I.ipynb delete mode 100644 experimental/Durham2024/material/Durham_2024_II.ipynb delete mode 100644 experimental/Durham2024/material/coord_imgs.mrdi diff --git a/experimental/Durham2024/material/Durham_2024_I.ipynb b/experimental/Durham2024/material/Durham_2024_I.ipynb deleted file mode 100644 index ce20d1867b70..000000000000 --- a/experimental/Durham2024/material/Durham_2024_I.ipynb +++ /dev/null @@ -1,641 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "id": "8da163a2", - "metadata": {}, - "source": [ - "# Algebraic Geometry in OSCAR: part I\n", - "\n", - "[OSCAR](https://www.computeralgebra.de) is a computer algebra system whose development is funded by the Deutsche Forschungsgemeinschaft (DFG) as part of the SFB-TRR 195. It is open source and can be found on [GitHub](www.github.com/oscar-system/Oscar.jl). The core functionality is for all kinds of *symbolic computations* and comprises number theory, group and representation theory, polyhedral geometry, commutative algebra, ...\n", - "\n", - "Today, we will have a look at how to use OSCAR to do Algebraic Geometry and, in particular, Intersection Theory. We will mostly follow **[Ful98]** and the [stacks project](https://stacks.math.columbia.edu/)." - ] - }, - { - "cell_type": "markdown", - "id": "a55e6e76", - "metadata": {}, - "source": [ - "### Installation and starting up OSCAR \n", - "\n", - "Installation instructions for OSCAR can be found on its [website](https://www.oscar-system.org/install/). \n", - "For the purpose of this course, we shall need a very particular branch of OSCAR which can be installed using the following commands." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "aa985b7c", - "metadata": {}, - "outputs": [], - "source": [ - "using Pkg\n", - "]add https://github.com/HechtiDerLachs/Oscar.jl#Durham_2024" - ] - }, - { - "cell_type": "markdown", - "id": "dede3a91", - "metadata": {}, - "source": [ - "To use OSCAR, start up `julia` and type" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "8aa0970f", - "metadata": {}, - "outputs": [], - "source": [ - "#using Revise, Oscar # Revise is another very useful, independent package \n", - " # which allows you to change code without restarting.\n", - "using Oscar" - ] - }, - { - "cell_type": "markdown", - "id": "5d5d4ffd", - "metadata": {}, - "source": [ - "You may need to do some updates, eventually:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "f7c0f6e9", - "metadata": {}, - "outputs": [], - "source": [ - "]up" - ] - }, - { - "cell_type": "markdown", - "id": "21fba4ed", - "metadata": {}, - "source": [ - "### First steps for Algebraic Geometry in OSCAR\n", - "\n", - "Let us define an elliptic curve $E$ over the function field $\\mathbb k(t)$ for $\\mathbb k = \\mathbb F_{113}$." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "2c260e95", - "metadata": {}, - "outputs": [], - "source": [ - "kk = GF(113) # the finite field (Galois field) with 113 elements" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "4327a12e", - "metadata": {}, - "outputs": [], - "source": [ - "kkt_poly, t_poly = kk[:t] # the polynomial ring in one variable `t` over `kk`" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "159c11a4", - "metadata": {}, - "outputs": [], - "source": [ - "kkt_field = fraction_field(kkt_poly) # the fraction field of that polynomial ring" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "0aa993cc", - "metadata": {}, - "outputs": [], - "source": [ - "t_field = kkt_field(t_poly) # the monomial `t`, but as an element of the fraction field" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "14367677", - "metadata": {}, - "outputs": [], - "source": [ - "P, (x, y) = kkt_field[:x, :y]; # the bivariate polynomial ring over `kkt_field`\n", - "P" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "0135afd3", - "metadata": {}, - "outputs": [], - "source": [ - "f = y^2 - x^3 - (t_poly^8 - 1)*x # a Weierstrass equation for an elliptic curve" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "549c5e73", - "metadata": {}, - "outputs": [], - "source": [ - "E = elliptic_curve(f, x, y) # the associated elliptic curve as an instance of `EllipticCurve`" - ] - }, - { - "cell_type": "markdown", - "id": "c6ca76fa", - "metadata": {}, - "source": [ - "### Affine schemes in OSCAR\n", - "\n", - "For an elliptic curve as above with *integral* coefficients, we can build the *Weierstrass model* of its associated *elliptic surface*." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "379dea9b", - "metadata": {}, - "outputs": [], - "source": [ - "P, (x, y, t) = kk[:x, :y, :t] # a multivariate polynomial ring in three variables\n", - "f = y^2 - x^3 - (t^8 - 1)*x # the defining equation for the Weierstrass chart\n", - "I = ideal(P, f) # a principal ideal in `P`" - ] - }, - { - "cell_type": "markdown", - "id": "bbf497c0", - "metadata": {}, - "source": [ - "To any finitely generated `kk`-algebra we can associate its [spectrum](https://stacks.math.columbia.edu/tag/01HR)." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "4709c18a", - "metadata": {}, - "outputs": [], - "source": [ - "IA3 = spec(P) # the affine scheme associated to `P`\n", - "W, inc_W = sub(IA3, I) # the closed subscheme defined by `I` together with its `ClosedEmbedding`\n", - "W" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "d770d898", - "metadata": {}, - "outputs": [], - "source": [ - "W_sing, inc_W_sing = singular_locus(W) # the singular locus of `W` with its `ClosedEmbedding`\n", - "W_sing" - ] - }, - { - "cell_type": "markdown", - "id": "2ce08ef2", - "metadata": {}, - "source": [ - "The second return value is a [closed embedding](https://stacks.math.columbia.edu/tag/01HJ) which has its own datatype and its own methods in OSCAR." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "f505834e", - "metadata": {}, - "outputs": [], - "source": [ - "typeof(inc_W_sing)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "53e3e75d", - "metadata": {}, - "outputs": [], - "source": [ - "methodswith(ClosedEmbedding)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "cb207814", - "metadata": {}, - "outputs": [], - "source": [ - "I_sing = image_ideal(inc_W_sing) # the ideal defining `W_sing` as a closed subscheme of `W`" - ] - }, - { - "cell_type": "markdown", - "id": "17583d78", - "metadata": {}, - "source": [ - "We can get our hands on the distict points of the singular locus via a primary decomposition." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "2a69a6f4", - "metadata": { - "scrolled": true - }, - "outputs": [], - "source": [ - "I_sing_dec = primary_decomposition(I_sing) # eight distinct points at `(x, y, t) = (0, 0, \\zeta_8^k)`" - ] - }, - { - "cell_type": "markdown", - "id": "63ed9eb3", - "metadata": {}, - "source": [ - "In OSCAR we can work with polynomial rings and their quotients, but also with *localizations*." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "5bf7175e", - "metadata": {}, - "outputs": [], - "source": [ - "U = hypersurface_complement(W, t^8-1)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "2a6a96a1", - "metadata": {}, - "outputs": [], - "source": [ - "Sigma, _ = singular_locus(U)\n", - "is_empty(Sigma)" - ] - }, - { - "cell_type": "markdown", - "id": "55f93291", - "metadata": {}, - "source": [ - "### Covered schemes in OSCAR\n", - "\n", - "The above were examples of *affine schemes* and their morphisms. These are but wrappers of objects in commutative algebra. But in order to construct the actual elliptic surface associated to $E$, we need to resolve the singularities of the Weierstrass model $W$. \n", - "\n", - "This will lead to *covered schemes* which are glued from affine schemes as for example described [here](https://stacks.math.columbia.edu/tag/01JA)." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "8ccc870e", - "metadata": {}, - "outputs": [], - "source": [ - "P1 = I_sing_dec[1][1] # an ideal in `OO(W)`\n", - "pr1 = blow_up(W, P1) # the projection morphism `W1 -> W`" - ] - }, - { - "cell_type": "markdown", - "id": "b1661ec9", - "metadata": {}, - "source": [ - "We have just [blown up](https://stacks.math.columbia.edu/tag/052P) the affine scheme `W` along the ideal `P1`." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "3b119926", - "metadata": {}, - "outputs": [], - "source": [ - "W1 = domain(pr1); # a `CoveredScheme` over `kk`\n", - "cov = default_covering(W1) # the initial `Covering` used to define the scheme" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "8b77f6d5", - "metadata": {}, - "outputs": [], - "source": [ - "U = patches(cov)[1] # an affine scheme; the first chart of `W1`" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "cdfde006", - "metadata": {}, - "outputs": [], - "source": [ - "V = patches(cov)[2]" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "0f47df3a", - "metadata": {}, - "outputs": [], - "source": [ - "glue = cov[U, V] # either two `patches` in a `Covering` might be glued along open subsets" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "872dc99d", - "metadata": {}, - "outputs": [], - "source": [ - "U_simp = simplify(U) # one can always try to eliminate variables using `simplify`" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "64edc87a", - "metadata": {}, - "outputs": [], - "source": [ - "E1 = exceptional_divisor(pr1) # the exceptional divisor of the blowup\n", - "typeof(E1)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "cede2895", - "metadata": {}, - "outputs": [], - "source": [ - "E1(U_simp) # the defining equations for the exceptional divisor in `U_simp`" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "faf7c210", - "metadata": {}, - "outputs": [], - "source": [ - "small_generating_set(ideal_sheaf(E1)(U)) # a glimpse of the ideal in the whole affine ambient space" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "3819cb0c", - "metadata": {}, - "outputs": [], - "source": [ - "is_prime(ideal_sheaf(E1)) # in this case the exceptional divisor is prime" - ] - }, - { - "cell_type": "markdown", - "id": "16140bbc", - "metadata": {}, - "source": [ - "We can perform (parts of) the usual operations for the blowup morphism `pr1`, such as e.g. pullback:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "2756772b", - "metadata": {}, - "outputs": [], - "source": [ - "W_cov = codomain(pr1) # a `CoveredScheme` with only one chart `W`\n", - "II = ideal_sheaf(W_cov, W, ideal(OO(W), t+1)) # the `AbsIdealSheaf` on `W_cov` for the fiber over `t = -1" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "ea66a2a6", - "metadata": {}, - "outputs": [], - "source": [ - "pb1_II = pullback(pr1, II) # the ideal sheaf for the fiber over `t = -1` in the blowup `W1`\n", - "pb1_II(U_simp)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "c431a7a6", - "metadata": {}, - "outputs": [], - "source": [ - "is_prime(pb1_II)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "75279b55", - "metadata": {}, - "outputs": [], - "source": [ - "F1 = weil_divisor(pb1_II) # the fiber as a `AbsWeilDivisor`\n", - "F1 = irreducible_decomposition(F1) # ...decomposed into irreducible components" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "002bb1da", - "metadata": {}, - "outputs": [], - "source": [ - "Q1, Q2 = components(F1);" - ] - }, - { - "cell_type": "markdown", - "id": "d72801d7", - "metadata": {}, - "source": [ - "Let us compare the components of the fiber with the exceptional divisor and the strict transform of the fiber." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "2b03afa4", - "metadata": {}, - "outputs": [], - "source": [ - "str1_II = strict_transform(pr1, II);\n", - "is_prime(str1_II)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "27e6b1e3", - "metadata": {}, - "outputs": [], - "source": [ - "ideal_sheaf(E1) == Q1" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "933d164d", - "metadata": {}, - "outputs": [], - "source": [ - "ideal_sheaf(E1) == Q2" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "d2fffd84", - "metadata": {}, - "outputs": [], - "source": [ - "str1_II == Q1" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "88698429", - "metadata": {}, - "outputs": [], - "source": [ - "str1_II == Q2" - ] - }, - { - "cell_type": "markdown", - "id": "c4057cd2", - "metadata": {}, - "source": [ - "For the fun of it, let us have a look at the `weil_divisor` associated to the square of the fiber ideal" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "2883a200", - "metadata": {}, - "outputs": [], - "source": [ - "JJ = pb1_II^2\n", - "@time irreducible_decomposition(weil_divisor(JJ))" - ] - }, - { - "cell_type": "markdown", - "id": "f1d22164", - "metadata": {}, - "source": [ - "How do the components of the fiber over `t = -1` intersect?" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "38dab652", - "metadata": {}, - "outputs": [], - "source": [ - "intersect(weil_divisor(Q1), weil_divisor(Q2)) # A custom method for Weil divisors on surfaces!" - ] - }, - { - "cell_type": "markdown", - "id": "98c9f73c", - "metadata": {}, - "source": [ - "How do we actually get this number?" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "46d6bdbb", - "metadata": {}, - "outputs": [], - "source": [ - "JJ = Q1 + Q2; # the sum of two ideal sheaves; will be 0-dimensional in this case\n", - "Z = AlgebraicCycle(W1, ZZ, IdDict([JJ=>one(ZZ)])) # the associated `AbsAlgebraicCycle` with ZZ-coefficients" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "e2bf0ef8", - "metadata": {}, - "outputs": [], - "source": [ - "irreducible_decomposition(Z) # decomposition into prime components with multiplicities" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "6d124b9d", - "metadata": {}, - "outputs": [], - "source": [ - "integral(ans) # pushforward to the point `spec(kk)`" - ] - }, - { - "cell_type": "markdown", - "id": "588793ae", - "metadata": {}, - "source": [ - "# References\n", - "\n", - " * **[Ful98]**: Fulton, W., Intersection Theory, Springer, 1998" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Julia 1.10.0", - "language": "julia", - "name": "julia-1.10" - }, - "language_info": { - "file_extension": ".jl", - "mimetype": "application/julia", - "name": "julia", - "version": "1.10.0" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/experimental/Durham2024/material/Durham_2024_II.ipynb b/experimental/Durham2024/material/Durham_2024_II.ipynb deleted file mode 100644 index 6ce7ab6a2cd5..000000000000 --- a/experimental/Durham2024/material/Durham_2024_II.ipynb +++ /dev/null @@ -1,818 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "id": "ed8afc8f", - "metadata": {}, - "source": [ - "# Algebraic Geometry in OSCAR: part II\n", - "\n", - "## A look into the OSCAR kitchen: What's cooking?\n", - "\n", - "Together with [Simon Brandhorst](https://www.math.uni-sb.de/ag/brandhorst/) we are currently working on finding an [automorphism of minimal entropy for Enriques surfaces](https://arxiv.org/pdf/1807.09452v2.pdf). To this end, we have implemented and automatized wide parts of the functionality for *elliptic surfaces*." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "c676d78b", - "metadata": {}, - "outputs": [], - "source": [ - "using Revise, Oscar" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "81757145", - "metadata": { - "scrolled": true - }, - "outputs": [], - "source": [ - "kk = GF(113) # the finite field (Galois field) with 113 elements\n", - "kkt_poly, t_poly = kk[:t] # the polynomial ring in one variable `t` over `kk`\n", - "kkt_field = fraction_field(kkt_poly) # the fraction field of that polynomial ring\n", - "P, (x, y) = kkt_field[:x, :y]; # the bivariate polynomial ring over `kkt_field`\n", - "f = y^2 - x^3 - (1 - t_poly^8)*x # a Weierstrass equation for an elliptic curve\n", - "E = elliptic_curve(f, x, y) # the associated elliptic curve as an instance of `EllipticCurve`" - ] - }, - { - "cell_type": "markdown", - "id": "c8cb97c0", - "metadata": {}, - "source": [ - "We can construct an elliptic surface directly from `E`." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "724727a5", - "metadata": {}, - "outputs": [], - "source": [ - "X = elliptic_surface(E, 2; resolution_strategy=:simultaneous)" - ] - }, - { - "cell_type": "markdown", - "id": "d2ff33ec", - "metadata": {}, - "source": [ - "Rational points on `E` will give rise to elements in the *Mordell-Weil lattice* of `X`. We have picked some points and make them known on `X`." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "6b0cef5f", - "metadata": {}, - "outputs": [], - "source": [ - "t = kkt_field(t_poly);\n", - "\n", - "basis_mwl = [[62*t^4 + 6*t^3 + 106*t^2 + 75*t + 26, 66*t^6 + 67*t^5 + 37*t^4 + 60*t^3 + 67*t^2 + 101*t + 58], [112*t^4 + 97*t^2 + 98, 104*t^5 + 82*t^3 + 91*t], [t^4 + 112, 87*t^4 + 26], [8*t^4 + 17*t^3 + 11*t^2 + 70*t + 7, 39*t^6 + 17*t^5 + 60*t^4 + 64*t^3 + 41*t^2 + 29*t + 89], [50*t^4 + 104*t^2 + 72, 56*t^6 + 30*t^4 + 36*t^2 + 104], [31*t^4 + 45*t^3 + 60*t^2 + 59*t + 13, 43*t^6 + 18*t^5 + 67*t^4 + 75*t^3 + 10*t^2 + 44*t + 17], [105*t^4 + 43*t^3 + 61*t^2 + 17*t + 7, 20*t^6 + 33*t^5 + 53*t^4 + 9*t^3 + 63*t^2 + 70*t + 89], [62*t^4 + 75*t^3 + 8*t^2 + 107*t + 87, 66*t^6 + 103*t^5 + 103*t^4 + 63*t^3 + 46*t^2 + 37*t + 34]]\n", - "\n", - "basis_mwl = [E(i) for i in basis_mwl]" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "0acb4fca", - "metadata": {}, - "outputs": [], - "source": [ - "Oscar.set_mordell_weil_basis!(X, basis_mwl)" - ] - }, - { - "cell_type": "markdown", - "id": "f3a68ae5", - "metadata": {}, - "source": [ - "In order to see more of what's going on, we may set up the verbosity." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "9a1030ef", - "metadata": {}, - "outputs": [], - "source": [ - "set_verbose_level(:EllipticSurface, 5)\n", - "set_verbose_level(:MorphismFromRationalFunctions, 5)" - ] - }, - { - "cell_type": "markdown", - "id": "22e56858", - "metadata": {}, - "source": [ - "For elliptic surfaces we can compute a basis for the **Neron-Severi group** whenever we have found a Mordell-Weil basis." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "d9f24377", - "metadata": { - "scrolled": true - }, - "outputs": [], - "source": [ - "# The following one command is only necessary for the correct bookkeeping with further code below; ignore this!\n", - "Oscar._trivial_lattice(X;reducible_singular_fibers_in_PP1=Oscar._reducible_fibers_disc(X;sort=false))\n", - "\n", - "# compute the full Neron-Severi lattice of `X`\n", - "alg_lat, _, NS = algebraic_lattice(X)\n", - "V = ambient_space(NS)" - ] - }, - { - "cell_type": "markdown", - "id": "2847fa4f", - "metadata": {}, - "source": [ - "The elements in `alg_lat` are divisors spanning the divisor class group up to *numerical equivalence*. \n", - "The big matrix is the *gram matrix* for their pairwise intersections." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "2e0eba30", - "metadata": { - "scrolled": true - }, - "outputs": [], - "source": [ - "alg_lat" - ] - }, - { - "cell_type": "markdown", - "id": "80ac901a", - "metadata": {}, - "source": [ - "Since `X` is a smooth, compact surface, the intersection of either two Weil divisors is well defined and we can compute pairwise intersection numbers." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "26cf3206", - "metadata": {}, - "outputs": [], - "source": [ - "intersect(alg_lat[1], alg_lat[12])" - ] - }, - { - "cell_type": "markdown", - "id": "35b118d7", - "metadata": {}, - "source": [ - "Even the self-intersection of all generators is known here! But unfortunately this is only known due to some tricks on elliptic surfaces..." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "070b85af", - "metadata": {}, - "outputs": [], - "source": [ - "intersect(alg_lat[1], alg_lat[1]) # the fiber of the elliptic fibration has self-intersection 0" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "2f2aca54", - "metadata": {}, - "outputs": [], - "source": [ - "intersect(alg_lat[3], alg_lat[3]) # any rational curve on a K3-surface has self intersection -2" - ] - }, - { - "cell_type": "markdown", - "id": "06c22c84", - "metadata": {}, - "source": [ - "### A dirty speedup for the self-intersection of curves\n", - "\n", - "> **Task 1:** Extend the computation of self-intersection of curves on a smooth surface by means of the [adjunction formula](https://en.wikipedia.org/wiki/Adjunction_formula) and the [Riemann-Roch theorem](https://en.wikipedia.org/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem) for curves.\n", - "\n", - "Roughly this can be done as follows.\n", - "\n", - "> **Theorem (Riemann-Roch for curves):** Let $C$ be a smooth projective curve and $D$ a divisor on it. Then \n", - "$$\\chi(D) = h^0(\\mathcal O(D)) - h^1(\\mathcal O(D)) = \\mathrm{deg}(D) + 1 - g(C)$$\n", - "where $g(C) = h^0(\\omega_C)$ is the *genus* and $\\mathrm{deg}(D) = \\int_C c_1(\\mathcal O(D))$ is the degree of the divisor.\n", - "\n", - "Applying this to the canonical divisor $D = \\omega_C$, we get the \n", - "well known formula $\\deg(\\omega_C) = 2g(C) - 2$. For the canonical bundle of $C$ we have a formula relating it to the canonical bundle of the ambient surface:\n", - "\n", - "> **Theorem (Adjunction formula):** Let $X$ be a smooth, projective surface and $i : C \\hookrightarrow X$ a smooth divisor on it. Then \n", - "$$ \\omega_C = i^* (\\omega_X \\otimes \\mathcal O(C)).$$\n", - "\n", - "Intersection theory tells us that $\\mathrm{deg}(i^*\\mathcal O(C)) = C.C$ is equal to the self-intersection of $C$ in $X$ and that the degree is logarithmic for line bundles on curves. Therefore \n", - "$$2g(C) - 2 = \\mathrm{deg}(\\omega_C) = \\mathrm{deg}(i^* \\omega_X) + \\deg(i^*\\mathcal O(C))) = K_X.C + C.C$$\n", - "where $K_X$ is the *canonical divisor* of the surface $X$.\n", - "\n", - "The algorithm would now proceed as follows:\n", - " * Is a canonical divisor of the surface known? If not, throw an error `can not compute self-intersection`.\n", - " * Is the curve smooth and in general position w.r.t $K_X$? If not, throw an error.\n", - " * Compute the genus of $C$.\n", - " * Return $2g(C) - 2 - C.K_X$.\n", - " \n", - "For the computation of the genus we may fall back to [implementations in Singular](https://www.singular.uni-kl.de/Manual/4-0-3/sing_1274.htm)." - ] - }, - { - "cell_type": "markdown", - "id": "25e400a2", - "metadata": {}, - "source": [ - "## Weil- and Cartier divisors\n", - "\n", - "> **Definition (algebraic cycle):** Let $X$ be a scheme. An *algebraic cycle* of dimension $k$ is a formal sum \n", - "$$ \\sum_{i} a_i \\cdot [V_i]$$\n", - "where $V_i \\subset X$ are $k$-dimensional *subvarieties* of $X$. \n", - "\n", - "Following **[Ful98]** we will only speak of a variety for reduced and irreducible schemes. The algebraic cycles of dimension $k$ form an Abelian group which will be denoted by $Z_k(X)$. \n", - "\n", - "> **Definition (Weil divisor):** Let $X$ be an equidimensional scheme of dimension $d$. A *Weil divisor* is an algebraic cycle on $X$ of dimension $d-1$.\n", - "\n", - "Some divisors are more special than others:\n", - "\n", - "> **Definition (Cartier divisor):** An *effective Cartier divisor* is given by an ideal sheaf $\\mathcal I$ on $X$ which is locally principal for some non-zero divisor. A *Cartier divisor* is a formal linear combination of effective Cartier divisors.\n", - "\n", - "In other words: For every point $p \\in X$ there exists an open neighborhood $U \\supset p$ and a non-zero divisor $f \\in \\mathcal O(U)$ such that $\\mathcal I(U) = \\langle f \\rangle \\subset \\mathcal O(U)$.\n", - "\n", - "For any two such ''local trivializations'' $(U, f)$ and $(V, g)$ of an effective Cartier divisor $C$ there is a unit $u \\in \\mathcal O(U \\cap V)$ such that $f|_{U \\cap V} = u \\cdot g|_{U \\cap V}$. These give rise to transition functions of a line bundle on $X$ which we will denote by $\\mathcal O(C)$. This extends naturally to all Cartier divisors by taking common refinements." - ] - }, - { - "cell_type": "markdown", - "id": "2d3548b6", - "metadata": {}, - "source": [ - "We also have a Cartier divisor floating around, coming as exceptional divisor of the resolution of singularities; but we first have to dig it out of the internals of the elliptic surface." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "43c96010", - "metadata": {}, - "outputs": [], - "source": [ - "Exc_amb = X.ambient_exceptionals[1] # a Cartier divisor on some smooth ambient space of `X`\n", - "Exc = pullback(X.inc_Y, Exc_amb) # the pullback of that divisor to `X` itself\n", - "first.(Exc.(affine_charts(X)))" - ] - }, - { - "cell_type": "markdown", - "id": "7a587975", - "metadata": {}, - "source": [ - "Any Cartier divisor gives rise to a Weil divisor:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "873b7c50", - "metadata": {}, - "outputs": [], - "source": [ - "Exc_weil = weil_divisor(Exc)\n", - "irreducible_decomposition(Exc_weil)" - ] - }, - { - "cell_type": "markdown", - "id": "7092773d", - "metadata": {}, - "source": [ - "The converse does not hold in general. However, it holds on *smooth* schemes and for every Weil divisor $D$ one may ask: ''Is $D$ actually Cartier?''\n", - "\n", - "> **Task 2:** Implement an algorithm which, given a Weil divisor $D$, decides whether $D$ is Cartier and, if so, computes a local trivialization for $D$.\n", - "\n", - "The point in this is, as we shall see later, that while the intersection of two Weil divisors is usually not well defined, the intersection of a Cartier- and a Weil divisor always is. In fact, Cartier divisors play a crucial role in the development of intersection theory." - ] - }, - { - "cell_type": "markdown", - "id": "c2237568", - "metadata": {}, - "source": [ - "### An automorphism of the K3 surface\n", - "\n", - "In our project with Simon Brandhorst we are constructing a very specific automorphism of a K3 surface (which is a lift to the universal cover of the Enriques surface automorphism proposed by Oguiso and Yu). K3 surfaces enjoy various favourable properties which allow us (more or less) to identify an automorphism $f \\colon X \\to X$ from its ''fingerprint'' given by the action on the Neron-Severi lattice\n", - "\n", - "$$ f_* \\colon \\mathrm{NS}(X) \\to \\mathrm{NS}(X).$$\n", - "\n", - "This should be thought of as something like a *hash function*: Any two given automorphisms $f$ and $g$ are very likely to agree if their actions on the Neron-Severi lattice do. However, given $f_*$ as above, it is almost impossible to reconstruct $X$ and $f$. This is basically what the whole paper with Simon is about!\n", - "\n", - "Let us have a look at the reconstruction of $f$." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "ab079123", - "metadata": {}, - "outputs": [], - "source": [ - "# the matrix representing the Oguiso-Yu automorphism w.r.t the basis of V\n", - "fOYonAlgX = QQ[5 1 0 -2 1 -1 0 -1 1 0 -1 0 -1 1 0 0 0 2; 2 -1 1 -2 2 0 1 -2 2 0 -2 0 -2 3 0 1 1 3; 4 1 0 -1 1 -1 0 -1 1 0 -1 0 -1 1 0 0 0 2; 5 1 0 -2 1 -1 0 -2 1 0 -1 0 -1 1 0 0 0 2; 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0; 5 1 0 -2 1 -1 0 -1 1 -1 -1 0 -1 1 0 0 0 2; 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 1 0 -2 1 0 0 -1 1 0 -1 0 -1 1 0 0 0 2; 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0; 10 -1 2 -7 5 -2 2 -4 5 1 -5 0 -5 7 0 2 2 9; 10 3 -1//2 -7//2 1//2 -3//2 -1//2 -5//2 1//2 -1//2 -2 0 -2 2 0 0 0 2; 7 1 1 -3 1 -1 0 -2 2 0 -3 0 -2 2 0 1 1 3; 4 0 1 -3 2 0 1 -3 1 1 -2 0 -3 3 1 1 0 3; 8 2 1 -3 1 -1 0 -2 2 0 -3 0 -2 2 -1 1 1 3; 10 0 2 -6 4 -2 1 -3 4 0 -5 0 -3 5 -1 2 2 7; 9 1 1//2 -9//2 5//2 -3//2 3//2 -7//2 5//2 1//2 -3 -1 -4 5 0 1 1 5; -3 -2 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1 1]" - ] - }, - { - "cell_type": "markdown", - "id": "0ba8189b", - "metadata": {}, - "source": [ - "So that is the fingerprint we need to realize. We have constructed a candidate for the automorphism; let's load it." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "ad29e9ac", - "metadata": {}, - "outputs": [], - "source": [ - "coord_imgs = load(\"coord_imgs.mrdi\")\n", - "coord_imgs = [evaluate(fr, gens(ambient_coordinate_ring(weierstrass_chart_on_minimal_model(X)))) for fr in coord_imgs]" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "693ffd41", - "metadata": {}, - "outputs": [], - "source": [ - "aut = morphism_from_rational_functions(X, X,\n", - " weierstrass_chart_on_minimal_model(X),\n", - " weierstrass_chart_on_minimal_model(X),\n", - " coord_imgs; check=false\n", - " );" - ] - }, - { - "cell_type": "markdown", - "id": "9e0b44e8", - "metadata": {}, - "source": [ - "Weil divisors are covariant in nature with respect to proper maps. \n", - "\n", - "> **Definition (proper pushforward):** Let $f \\colon X \\to Y$ be a proper morphism of algebraic schemes, $P \\in X$ a point of dimension $p$ and $Q = f(P) \\in Y$ its image. The pushforward of the cycle $[P] \\in Z_p(X)$ is defined as \n", - "$$ f_*([P]) = \\begin{cases} [R(P):R(Q)] \\cdot [Q] &\\mathrm{ if } \\quad \\dim Q = \\dim P \\\\ 0 & \\mathrm{ otherwise.} \\end{cases}$$\n", - "The general pushforward for algebraic cycles is defined by linear extension to $Z_p(X)$.\n", - "\n", - "Here we denote by $R(P)$ the field of rational functions on the subvariety of $X$ given by $P$ and vice versa for $Q$. \n", - "\n", - "For an *isomorphism* we can skip the computation of the degree of field extensions and just compute the image of a point." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "430bc097", - "metadata": { - "scrolled": true - }, - "outputs": [], - "source": [ - "set_attribute!(aut, :is_isomorphism=>true)\n", - "D = alg_lat[3]\n", - "pf_D = pushforward(aut, D)" - ] - }, - { - "cell_type": "markdown", - "id": "0b6b4f7a", - "metadata": {}, - "source": [ - "> **Task 3:** Extend the functionality for pushforward of algebraic cycles to more than just isomorphisms.\n", - "\n", - "One possibility would, for instance, be blowdown maps where we know that they are proper and birational. More generally, we do not have a good method to compute the degree of the field extension directly. But one could certainly think of one or two dirty workarounds for this..." - ] - }, - { - "cell_type": "markdown", - "id": "af7ce9c2", - "metadata": {}, - "source": [ - "For K3 surfaces we can do more than just push forward a divisor: We can represent the image divisor in our chosen basis for the Neron-Severi lattice!" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "ab332441", - "metadata": {}, - "outputs": [], - "source": [ - "rep_pf_D = basis_representation(X, pf_D)" - ] - }, - { - "cell_type": "markdown", - "id": "4d33ea45", - "metadata": {}, - "source": [ - "For elliptically fibered K3 surfaces we have an automated and optimized method to do everything at once." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "4e7f951d", - "metadata": {}, - "outputs": [], - "source": [ - "pf_aut = Oscar.pushforward_on_algebraic_lattices(aut) # CAUTION: This command takes some time for computation!" - ] - }, - { - "cell_type": "markdown", - "id": "116c7f3a", - "metadata": {}, - "source": [ - "That finding a basis representation for a divisor actually works is due to a rather remarkable property of K3 surfaces: The notions of rational-, algebraic-, and numerical equivalence agree. Consequently, we can compute a basis representation for any divisor from its intersection numbers with the generators of the Neron-Severi lattice.\n", - "\n", - "Note, however, that for a general surface it is almost impossible to determine whether or not two given divisors are even rationally equivalent!\n", - "\n", - "Before we move on, let us once compare the induced map on $\\mathrm{NS}(X)$ for the given automorphism `aut` with the fingerprint of `f`:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "e0a1e098", - "metadata": {}, - "outputs": [], - "source": [ - "matrix(pf_aut)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "386e439a", - "metadata": {}, - "outputs": [], - "source": [ - "fOYonAlgX" - ] - }, - { - "cell_type": "markdown", - "id": "cb33f6ad", - "metadata": {}, - "source": [ - "As you can see, we did not get completely lucky this time. Some more adjustments seem to be needed." - ] - }, - { - "cell_type": "markdown", - "id": "b5215c37", - "metadata": {}, - "source": [ - "### A brief review of rational equivalence\n", - "\n", - "Let $X$ be a scheme, $U \\subset X$ an affine open subset, and $f \\in \\mathcal O(U)$ a function. \n", - "For a prime ideal $P \\subset \\mathcal O(U)$ of codimension one we define the *order* of $f$ at \n", - "the point $P$ as \n", - "\n", - "$$ \\mathrm{ord}_P(f) = \\mathrm{length}(\\mathcal O_{X, P}/\\langle f\\rangle)$$\n", - "\n", - "where $\\mathcal O_{X, P} = \\mathcal O(U)_P$ is the localization at the complement of $P$.\n", - "This ring is local of dimension one, so that the quotient by $\\langle h\\rangle$ is zero-dimensional and hence has finite length.\n", - "\n", - "\n", - "If $X$ is a variety, then this extends naturally to rational functions on $X$ and we get the following notion.\n", - "\n", - "> **Definition (principal divisor):** Let $W$ be a variety and $f \\in R(W)$ a rational function. The *principal divisor* associated to $f$ is \n", - "$$\\mathrm{div}(f) = \\sum_{P \\subset W} \\mathrm{ord}_P(f) \\cdot [P].$$\n", - "\n", - "Moreover, two algebraic cycles can be related by the following equivalence.\n", - "\n", - "> **Definition (rational equivalence):** Two algebraic cycles $D, E \\in Z_k(X)$ on a scheme $X$ are called *rationally equivalent* $D \\sim E$ if there exist a finite number of subvarieties $W_i \\subset X$ of dimension $k+1$ and rational functions $f_i \\in R(W_i)$ such that \n", - "$$ D - E = \\sum_i \\mathrm{div}(f_i).$$\n", - "\n", - "The quotient $Z_k(X)/\\sim$ will be denoted by $A_k(X)$, the $k$-th *Chow group* of $X$.\n", - "\n", - "> **Proposition (pushforward on Chow groups):** The pushforward for proper maps descends to a morphism of Chow groups.\n", - "\n", - "See e.g. **[Ful98]**, Theorem 1.4." - ] - }, - { - "cell_type": "markdown", - "id": "5072cc58", - "metadata": {}, - "source": [ - "We do already have a data type for rational functions on varieties:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "72fd6873", - "metadata": {}, - "outputs": [], - "source": [ - "RX = function_field(X)" - ] - }, - { - "cell_type": "markdown", - "id": "7c021124", - "metadata": {}, - "source": [ - "Elements of this field are represented on the different charts `U` of `X` as fractions of polynomials for the `ambient_coordinate_ring` of `U`. This is mostly due to the fact that we have no implementation of fraction fields for quotients of polynomial rings at this time. " - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "3aee41f5", - "metadata": {}, - "outputs": [], - "source": [ - "W = weierstrass_chart_on_minimal_model(X)\n", - "R = ambient_coordinate_ring(W)\n", - "(x, y, t) = gens(R)\n", - "f = RX(x) # x as a rational function on `X`; potentially with poles outside `W`" - ] - }, - { - "cell_type": "markdown", - "id": "24c660cc", - "metadata": {}, - "source": [ - "Let us look at a representative of $f$ in another chart:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "e067eaf5", - "metadata": {}, - "outputs": [], - "source": [ - "simp_cov = Oscar.simplified_covering(X)\n", - "U = simp_cov[5]\n", - "f[U] # Note that we are using rectangular brackets for this at the moment!" - ] - }, - { - "cell_type": "markdown", - "id": "82e96ac0", - "metadata": {}, - "source": [ - "Note that representatives are not necessarily automatically reduced." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "be1f453d", - "metadata": {}, - "outputs": [], - "source": [ - "g = RX(first(gens(modulus(OO(W)))))\n", - "g[U]" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "7586ae90", - "metadata": {}, - "outputs": [], - "source": [ - "is_zero(g)" - ] - }, - { - "cell_type": "markdown", - "id": "4e862dcf", - "metadata": {}, - "source": [ - "Determining the principal divisor associated to a rational function is something which is actually still missing from our implementations!\n", - "\n", - "> **Task 4:** Implement an algorithm which computes $\\mathrm{div}(f)$ for a rational function $f \\in R(X)$ on a variety $X$." - ] - }, - { - "cell_type": "markdown", - "id": "cf973d9f", - "metadata": {}, - "source": [ - "However, we already have a method to determine the order of vanishing of a rational function on a divisor which might be helpful." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "0ed0ca2e", - "metadata": {}, - "outputs": [], - "source": [ - "I = ideal(OO(W), y)\n", - "decomp = primary_decomposition(I)\n", - "P = decomp[1][1]" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "1e8d1cf7", - "metadata": {}, - "outputs": [], - "source": [ - "D = weil_divisor(ideal_sheaf(X, W, P))\n", - "order_of_vanishing(f, D)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "e5deb4e7", - "metadata": { - "scrolled": true - }, - "outputs": [], - "source": [ - "order_of_vanishing(f^(-2), D)" - ] - }, - { - "cell_type": "markdown", - "id": "61e1bdc5", - "metadata": {}, - "source": [ - "## More on Cartier divisors\n", - "\n", - "While Weil divisors are of homological nature (i.e. they have a covariant behaviour), Cartier divisors are cohomological: Given a morphism of schemes $f \\colon X \\to Y$ and a Cartier divisor $C$ on $Y$ there should be a natural notion of ''pulling back'' $C$. Unfortunately, this is not quite the case.\n", - "\n", - "### Cartier divisors on varieties\n", - "On a *variety* $X$ we can define Cartier divisors to be given by a collection $(U_i, f_i)$ where $f_i \\in R(X)$ is a rational function and for any pairs $U_i \\cap U_j$ the quotient $f_i/f_j \\in \\mathcal O(U_i \\cap U_j)^*$.\n", - "In theory, this spares us the detour to complete the monoid of effective Cartier divisors.\n", - "\n", - "Recall that to each Cartier divisor $C$ we can associate a line bundle $\\mathcal L = \\mathcal O(C)$. \n", - "A rational function $f \\in R(X)$ determines a *principal* Cartier divisor and we may call two Cartier divisors \n", - "*rationally equivalent*, if their difference is principal. \n", - "\n", - "> **Lemma:** On a variety $X$ there is a $1:1$-correspondence between Cartier divisors up to rational equivalence and isomorphism classes of line bundles. \n", - "\n", - "In order to get back a Cartier divisor from a line bundle $\\mathcal L$ we may construct a *rational section* $s$ of $\\mathcal L$: Start on one chart $U_0$ of a local trivialization of $\\mathcal L$ and extend the local algebraic section $e$ for this trivialization as a rational section to all of $X$. In the notation above, the associated Cartier divisor will then be given by the collection $(U_j, f_j/f_0)$. We denote it by $c_1(\\mathcal L)$. \n", - "\n", - "### Intersection of a Cartier divisor with an algebraic cycle\n", - "\n", - "Let $X$ be a scheme and $D = \\sum_i a_i \\cdot W_i \\in Z_k(X)$ an algebraic cycle with $W_i$ irreducible. For a Cartier divisor $C = \\{(U_j, f_j)\\}_{j=1}^N$ we may define the intersection $C \\cap D$ as follows.\n", - "\n", - " * We reduce to the case of irreducible cycles by $C \\cap (\\sum_i a_i \\cdot W_i) = \\sum_i a_i \\cdot (C \\cap W_i)$.\n", - " * For an irreducible cycle $\\iota \\colon W \\hookrightarrow X$ we set \n", - " $$C \\cap W := \\iota_*(c_1(\\iota^* \\mathcal O(C))\\cap [W]).$$\n", - " * $\\iota^*\\mathcal O(C)$ is the pullback of the line bundle $\\mathcal O(C)$ associated to $C$. \n", - " * $\\iota_*$ denotes the proper pushforward along the closed embedding $\\iota$.\n", - " * $[W]$ denotes the fundamental class of $[W]$. \n", - " * $c_1(\\mathcal L) \\cap [W]$ is the Weil divisor on $W$ associated to the Cartier divisor $c_1(\\mathcal L)$." - ] - }, - { - "cell_type": "markdown", - "id": "1d981f7f", - "metadata": {}, - "source": [ - "### More general intersections of divisors on surfaces\n", - "\n", - "The current implementation has a method `intersect(D::AbsAlgebraicCycle, C::CartierDivisor)`, but it is not fully implemented yet! Let's have a look at some examples." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "f06621ab", - "metadata": { - "scrolled": false - }, - "outputs": [], - "source": [ - "Z = intersect(alg_lat[1], Exc)\n", - "integral(Z)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "3867e729", - "metadata": {}, - "outputs": [], - "source": [ - "Z = intersect(alg_lat[2], Exc)\n", - "integral(Z)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "66b77bae", - "metadata": {}, - "outputs": [], - "source": [ - "Z = intersect(alg_lat[3], Exc)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "01dada74", - "metadata": {}, - "outputs": [], - "source": [ - "Z = intersect(alg_lat[12], Exc)\n", - "integral(Z)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "b10822bb", - "metadata": {}, - "outputs": [], - "source": [ - "Z_decomp = irreducible_decomposition(Z)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "f960a263", - "metadata": {}, - "outputs": [], - "source": [ - "integral(ans)" - ] - }, - { - "cell_type": "markdown", - "id": "f0dcbb90", - "metadata": {}, - "source": [ - "This leads us to some more tricky tasks to achieve this week.\n", - "\n", - "> **Task 5:** Fill the gaps in the implementation of `intersect` for an `EffectiveCartierDivisor` and an `AbsAlgebraicCycle`.\n", - "\n", - "**Warning:** This is a deep dive! We will need some preparation for that.\n", - "\n", - "> **Task 6:** Speed up the implementation for `irreducible_decomposition`. \n", - "\n", - "It seems that a lot of time is spent in the generic method. But for many useful applications it should be possible to use more efficient shortcuts!\n", - "\n", - "> **Task 7:** Implement a method to construct the line bundle $\\mathcal O(C)$ associated to a Cartier divisor.\n", - "\n", - "This is rather technical. 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\ No newline at end of file From 1df0aef8fc57fd99f4e498321cf86e2d1cdb9d54 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 14:35:22 +0100 Subject: [PATCH 25/41] Move methods and fixes to appropriate places. --- experimental/Durham2024/src/fixes.jl | 190 ---------------- experimental/Durham2024/test/intersections.jl | 4 +- .../Schemes/Divisors/AlgebraicCycles.jl | 19 ++ .../Schemes/Divisors/CartierDivisor.jl | 4 + .../Schemes/Divisors/WeilDivisor.jl | 205 ++++++++++++++++++ .../Schemes/FunctionField/FunctionFields.jl | 3 + .../Schemes/Sheaves/IdealSheaves.jl | 9 + 7 files changed, 242 insertions(+), 192 deletions(-) diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl index 75231033aceb..d0360c1b7a27 100644 --- a/experimental/Durham2024/src/fixes.jl +++ b/experimental/Durham2024/src/fixes.jl @@ -1,21 +1,3 @@ -function components(::Type{T}, D::AbsAlgebraicCycle) where {T <: AbsAlgebraicCycle} - X = scheme(D) - R = coefficient_ring(D) - return [AlgebraicCycle(X, R, IdDict{AbsIdealSheaf, elem_type(R)}([I=>one(R)]); check=false) for I in components(D)] -end - -function components(::Type{T}, D::AbsWeilDivisor) where {T <: AbsWeilDivisor} - X = scheme(D) - R = coefficient_ring(D) - return [WeilDivisor(X, R, IdDict{AbsIdealSheaf, elem_type(R)}([I=>one(R)]); check=false) for I in components(D)] -end - -function getindex(D::AbsAlgebraicCycle, C::AbsAlgebraicCycle) - comps = components(C) - @assert isone(length(comps)) - return D[first(comps)] -end - function canonical_divisor(X:: AbsAffineScheme{<:Field, <:MPolyRing}) return CartierDivisor(covered_scheme(X), ZZ) end @@ -27,175 +9,3 @@ function self_intersection_via_adjunction(K_X::CartierDivisor, C::CartierDivisor return 2g-2-sum(a*intersect(K_X, comp) for (a, comp) in coefficient_dict(C); init=0) end - - - -scheme(f::VarietyFunctionFieldElem) = scheme(parent(f)) -variety(f::VarietyFunctionFieldElem) = variety(parent(f)) - -# Compute the colength of I_P in the localization R_P. -# This assumes that R itself is a domain and that I is of height 1. -# Then there exists a regular -# point on Spec(R) and hence R_P is also regular and a UFD. -# Being of height 1, I_P must then be principal. -function _colength_in_localization(I::Ideal, P::Ideal) - R = base_ring(I) - @assert R === base_ring(P) - U = MPolyComplementOfPrimeIdeal(saturated_ideal(P)) - L, loc = localization(R, U) - I_loc = loc(I) - @assert base_ring(I_loc) === L - P_loc = loc(P) - x = _find_principal_generator(P_loc) - y = one(L) - k = 0 - while true - (y in I_loc) && return k - y = y*x - k += 1 - end - - return _colength_in_localization(saturated_ideal(I), saturated_ideal(P)) -end - -# same assumptions as above apply. -function _find_principal_generator(I::Union{<:MPolyLocalizedIdeal, <:MPolyQuoLocalizedIdeal}) - L = base_ring(I) - g = gens(I) - g = sort!(g, by=x->total_degree(lifted_numerator(x))) - for x in g - is_zero(x) && continue - ideal(L, x) == I && return x - end - error("no principal generator found") -end - -function weil_divisor( - f::VarietyFunctionFieldElem; - ring::Ring=ZZ, covering::Covering=default_covering(scheme(f)) - ) - @vprint :Divisors 4 "calculating principal divisor for $f\n" - X = scheme(f) - ideal_dict = IdDict{AbsIdealSheaf, elem_type(ring)}() - for U in patches(covering) - @vprint :Divisors 4 "doing patch $U\n" - inc_dict = IdDict{AbsIdealSheaf, elem_type(ring)}() - f_loc = f[U] - num = numerator(f_loc) - den = denominator(f_loc) - num_ideal = ideal(OO(U), num) - den_ideal = ideal(OO(U), den) - num_dec = primary_decomposition(num_ideal) - den_dec = primary_decomposition(den_ideal) - @vprint :Divisors 4 " numerator:\n" - for (_, P) in num_dec - @vprint :Divisors 4 " $P\n" - # If this component was already seen in another patch, skip it. - new_comp = PrimeIdealSheafFromChart(X, U, P) - @vprint :Divisors 4 " $(any(new_comp == PP for PP in keys(ideal_dict)) ? "already found" : "new component")\n" - any(new_comp == PP for PP in keys(ideal_dict)) && continue - c = _colength_in_localization(num_ideal, P) - @vprint :Divisors 4 " multiplicity $c\n" - inc_dict[new_comp] = c - end - @vprint :Divisors 4 " denominator:\n" - for (_, P) in den_dec - # If this component was already seen in another patch, skip it. - new_comp = PrimeIdealSheafFromChart(X, U, P) - @vprint :Divisors 4 " $(any(new_comp == PP for PP in keys(ideal_dict)) ? "already found" : "new component")\n" - any(new_comp == PP for PP in keys(ideal_dict)) && continue - c = _colength_in_localization(den_ideal, P) - @vprint :Divisors 4 " multiplicity $c\n" - key_list = collect(keys(inc_dict)) - k = findfirst(==(new_comp), key_list) - if k === nothing - is_zero(c) && continue - inc_dict[new_comp] = -c - else - d = inc_dict[key_list[k]] - if c == d - delete!(inc_dict, key_list[k]) - continue - end - inc_dict[key_list[k]] = d - c - end - end - for (pp, c) in inc_dict - ideal_dict[pp] = c - end - end - return WeilDivisor(X, ring, ideal_dict; check=false) -end - -function move_divisor(D::AbsWeilDivisor; check::Bool=false) - X = scheme(D) - is_zero(D) && return D - - if !is_prime(D) - R = coefficient_ring(D) - return sum(a*move_divisor(WeilDivisor(D, R; check=false)) for (D, a) in coefficient_dict(irreducible_decomposition(D)); init=WeilDivisor(X, R)) - end - - P = first(components(D)) - i = findfirst(U->!isone(P(U)), affine_charts(X)) - i === nothing && error("divisor is trivial") - U = affine_charts(X)[i] - I = P(U) - L, loc = localization(OO(U), complement_of_prime_ideal(I)) - LP = loc(I) - g = gens(saturated_ideal(I)) - g = sort!(g, by=total_degree) - i = findfirst(f->(ideal(L, f) == LP), g) - x = g[i] - f = function_field(X; check)(x) - return irreducible_decomposition(D - weil_divisor(f)) -end - -function is_zero(D::AbsAlgebraicCycle) - return all(is_zero(c) || is_one(I) for (I, c) in coefficient_dict(D)) -end - -function intersect(C::CartierDivisor, D::AbsWeilDivisor) - X = scheme(C) - @assert X === scheme(D) - R = coefficient_ring(C) - @assert R === coefficient_ring(D) - result = AlgebraicCycle(X, R) - for (E, c) in coefficient_dict(C) - result = result + c*intersect(E, D) - end - return result -end - -function intersect(E::EffectiveCartierDivisor, D::AbsWeilDivisor; check::Bool=true) - X = scheme(E) - @assert X === scheme(D) - R = coefficient_ring(D) - result = AlgebraicCycle(X, R) - cpcd = copy(coefficient_dict(D)) - DD = irreducible_decomposition(D) - cpcd = copy(coefficient_dict(DD)) - for (P, a) in coefficient_dict(DD) - _, inc_P = sub(P) - if is_zero(pullback(inc_P, ideal_sheaf(E))) - P_moved = move_divisor(WeilDivisor(X, R, - IdDict{AbsIdealSheaf, elem_type(R)}( - [P=>one(R)]); check=false); check=false) - result = result + a*intersect(E, P_moved) - else - result = result + a*AlgebraicCycle(X, R, - IdDict{AbsIdealSheaf, elem_type(R)}( - [P + ideal_sheaf(E)=>one(R)]); check=false) - end - end - return result -end - -function is_zero(II::AbsIdealSheaf) - return all(iszero(II(U)) for U in affine_charts(scheme(II))) -end - -function -(C::CartierDivisor, E::EffectiveCartierDivisor) - return C - 1*E -end - diff --git a/experimental/Durham2024/test/intersections.jl b/experimental/Durham2024/test/intersections.jl index f370a48bd760..c9943944fb25 100644 --- a/experimental/Durham2024/test/intersections.jl +++ b/experimental/Durham2024/test/intersections.jl @@ -86,9 +86,9 @@ integral(Int123); E2_sub, inc_E2 = sub(ideal_sheaf(E2_final)) pb_E1 = pullback(inc_E2, E1_final) pb_E1_weil = weil_divisor(pb_E1) -int1 = integral(intersect(pb_E1, pb_E1_weil)) +int1 = integral(Oscar._intersect(pb_E1, pb_E1_weil)) move = Oscar.move_divisor(pb_E1_weil) -int2 = integral(intersect(pb_E1, move)) +int2 = integral(Oscar._intersect(pb_E1, move)) @test int1 == int2 diff --git a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl index 74b4db60e52e..607c229d08aa 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl @@ -558,3 +558,22 @@ function integral(W::AbsAlgebraicCycle; check::Bool=true) return result end +# Getters for the components as honest algebraic cycles, not ideal sheaves. +function components(::Type{T}, D::AbsAlgebraicCycle) where {T <: AbsAlgebraicCycle} + X = scheme(D) + R = coefficient_ring(D) + return [AlgebraicCycle(X, R, IdDict{AbsIdealSheaf, elem_type(R)}([I=>one(R)]); check=false) for I in components(D)] +end + +function components(::Type{T}, D::AbsWeilDivisor) where {T <: AbsWeilDivisor} + X = scheme(D) + R = coefficient_ring(D) + return [WeilDivisor(X, R, IdDict{AbsIdealSheaf, elem_type(R)}([I=>one(R)]); check=false) for I in components(D)] +end + +function getindex(D::AbsAlgebraicCycle, C::AbsAlgebraicCycle) + comps = components(C) + @assert isone(length(comps)) + return D[first(comps)] +end + diff --git a/src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl index da480dc40c9b..bac22c453863 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl @@ -88,6 +88,10 @@ function +(C::CartierDivisor, D::CartierDivisor) return CartierDivisor(ambient_scheme(C), coefficient_ring(C), coeff_dict) end +function -(C::CartierDivisor, E::EffectiveCartierDivisor) + return C - 1*E +end + function +(C::CartierDivisor, D::EffectiveCartierDivisor) return C + CartierDivisor(D) end diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index 33a5ce98411e..0ba27e139e80 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -630,3 +630,208 @@ function order_of_vanishing( I = components(D)[1] return order_of_vanishing(f, I, check=false) end + +# Compute the colength of I_P in the localization R_P. +# This assumes that R itself is a domain and that I is of height 1. +# Then there exists a regular +# point on Spec(R) and hence R_P is also regular and a UFD. +# Being of height 1, I_P must then be principal. +function _colength_in_localization(I::Ideal, P::Ideal) + R = base_ring(I) + @assert R === base_ring(P) + U = MPolyComplementOfPrimeIdeal(saturated_ideal(P); check=false) + L, loc = localization(R, U) + I_loc = loc(I) + @assert base_ring(I_loc) === L + P_loc = loc(P) + x = _find_principal_generator(P_loc) + y = one(L) + k = 0 + while true + (y in I_loc) && return k + y = y*x + k += 1 + end + + return _colength_in_localization(saturated_ideal(I), saturated_ideal(P)) +end + +# same assumptions as above apply. +function _find_principal_generator(I::Union{<:MPolyLocalizedIdeal, <:MPolyQuoLocalizedIdeal}) + L = base_ring(I) + g = gens(I) + g = sort!(g, by=x->total_degree(lifted_numerator(x))) + for x in g + is_zero(x) && continue + ideal(L, x) == I && return x + end + error("no principal generator found") +end + +# produce the principal divisor associated to a rational function +function weil_divisor( + f::VarietyFunctionFieldElem; + ring::Ring=ZZ, covering::Covering=default_covering(scheme(f)) + ) + @vprint :Divisors 4 "calculating principal divisor for $f\n" + X = scheme(f) + ideal_dict = IdDict{AbsIdealSheaf, elem_type(ring)}() + for U in patches(covering) + @vprint :Divisors 4 "doing patch $U\n" + inc_dict = IdDict{AbsIdealSheaf, elem_type(ring)}() + f_loc = f[U] + num = numerator(f_loc) + den = denominator(f_loc) + num_ideal = ideal(OO(U), num) + den_ideal = ideal(OO(U), den) + num_dec = primary_decomposition(num_ideal) + den_dec = primary_decomposition(den_ideal) + @vprint :Divisors 4 " numerator:\n" + for (_, P) in num_dec + @vprint :Divisors 4 " $P\n" + # If this component was already seen in another patch, skip it. + new_comp = PrimeIdealSheafFromChart(X, U, P) + @vprint :Divisors 4 " $(any(new_comp == PP for PP in keys(ideal_dict)) ? "already found" : "new component")\n" + any(new_comp == PP for PP in keys(ideal_dict)) && continue + c = _colength_in_localization(num_ideal, P) + @vprint :Divisors 4 " multiplicity $c\n" + inc_dict[new_comp] = c + end + @vprint :Divisors 4 " denominator:\n" + for (_, P) in den_dec + # If this component was already seen in another patch, skip it. + new_comp = PrimeIdealSheafFromChart(X, U, P) + @vprint :Divisors 4 " $(any(new_comp == PP for PP in keys(ideal_dict)) ? "already found" : "new component")\n" + any(new_comp == PP for PP in keys(ideal_dict)) && continue + c = _colength_in_localization(den_ideal, P) + @vprint :Divisors 4 " multiplicity $c\n" + key_list = collect(keys(inc_dict)) + k = findfirst(==(new_comp), key_list) + if k === nothing + is_zero(c) && continue + inc_dict[new_comp] = -c + else + d = inc_dict[key_list[k]] + if c == d + delete!(inc_dict, key_list[k]) + continue + end + inc_dict[key_list[k]] = d - c + end + end + for (pp, c) in inc_dict + ideal_dict[pp] = c + end + end + return WeilDivisor(X, ring, ideal_dict; check=false) +end + +@doc raw""" + move_divisor(D::AbsWeilDivisor; check::Bool=false) + +Given an `AbsWeilDivisor` `D` on a scheme `X`, create a principal divisor +`div(f)` for some rational function, so that `D - div(f)` is supported +away from the original support of `D`. + +Keyword arguments: + * `randomization`: By default, the choices made to create `f` keep it as simple as possible. However, one might encounter constellations where this will just swap two components of `D`. In order to avoid this, one can then switch on randomization here. + * `is_prime`: Set this to `true` if you know your divisor `D` to already be prime and avoid expensive internal checks. +""" +function move_divisor( + D::AbsWeilDivisor; + check::Bool=false, + randomization::Bool=false, + is_prime::Bool=false + ) + X = scheme(D) + @check is_irreducible(X) && is_reduced(X) "scheme must be irreducible and reduced" + is_zero(D) && return D + + if !is_prime && !Oscar.is_prime(D) + R = coefficient_ring(D) + return sum(a*move_divisor(WeilDivisor(D, R; check=false)) for (D, a) in coefficient_dict(irreducible_decomposition(D)); init=WeilDivisor(X, R)) + end + + # We may assume that `D` is prime. + P = first(components(D)) + # find a chart where the support is visible + i = findfirst(U->!isone(P(U)), affine_charts(X)) + i === nothing && error("divisor is trivial") + U = affine_charts(X)[i] + I = P(U) + L, loc = localization(OO(U), complement_of_prime_ideal(I)) + LP = loc(I) + # Find a principal generator for the local ideal. + # This works, because we assume that `X` is irreducible and reduced. + # Then there is at least one smooth point on `U` and therefore `LP` + # is regular. Regular domains are UFD and an ideal of height 1 is + # principal there. + g = gens(saturated_ideal(I)) + g = sort!(g, by=total_degree) + i = findfirst(f->(ideal(L, f) == LP), g) + x = g[i] + f = function_field(X; check)(x) + if randomization + kk = base_ring(X) + R = ambient_coordinate_ring(U) + y = rand(kk, 1:10) + sum(rand(kk, 1:10)*a for a in gens(R); init=zero(R)) + f = f*inv(parent(f)(y)) + end + result = irreducible_decomposition(D - weil_divisor(f)) + # Check whether the supports are really different + if any(any(P == Q for Q in components(D)) for P in components(result)) + @show "switched component" + return move_divisor(D; randomization=true, check, is_prime) + end + return result +end + +function is_zero(D::AbsAlgebraicCycle) + return all(is_zero(c) || is_one(I) for (I, c) in coefficient_dict(D)) +end + +# Internal method which performs an automated move of the second argument +# if things are not in sufficiently general position. +# The problem is: We have no way to check whether a variety is proper over +# its base field. But the output is only valid if that is true. +# Therefore, we have no choice, but to hide it from the user for now. +function _intersect(C::CartierDivisor, D::AbsWeilDivisor) + X = scheme(C) + @assert X === scheme(D) + R = coefficient_ring(C) + @assert R === coefficient_ring(D) + result = AlgebraicCycle(X, R) + for (E, c) in coefficient_dict(C) + result = result + c*_intersect(E, D) + end + return result +end + +function _intersect(E::EffectiveCartierDivisor, D::AbsWeilDivisor; check::Bool=true) + X = scheme(E) + @assert X === scheme(D) + R = coefficient_ring(D) + result = AlgebraicCycle(X, R) + cpcd = copy(coefficient_dict(D)) + DD = irreducible_decomposition(D) + cpcd = copy(coefficient_dict(DD)) + for (P, a) in coefficient_dict(DD) + _, inc_P = sub(P) + # WARNING: The heuristic implemented below might still run into an infinite loop! + # We have to think about how this can effectively be avoided. See the test file + # for an example. + if is_zero(pullback(inc_P, ideal_sheaf(E))) + P_moved = move_divisor(WeilDivisor(X, R, + IdDict{AbsIdealSheaf, elem_type(R)}( + [P=>one(R)]); check=false); check=false, randomization=true, is_prime=true) + result = result + a*_intersect(E, P_moved) + else + result = result + a*AlgebraicCycle(X, R, + IdDict{AbsIdealSheaf, elem_type(R)}( + [P + ideal_sheaf(E)=>one(R)]); check=false) + end + end + return result +end + + diff --git a/src/AlgebraicGeometry/Schemes/FunctionField/FunctionFields.jl b/src/AlgebraicGeometry/Schemes/FunctionField/FunctionFields.jl index 91c018e0b6e0..710e252f4d97 100644 --- a/src/AlgebraicGeometry/Schemes/FunctionField/FunctionFields.jl +++ b/src/AlgebraicGeometry/Schemes/FunctionField/FunctionFields.jl @@ -517,3 +517,6 @@ function pullback(f::AbsCoveredSchemeMorphism, a::VarietyFunctionFieldElem) end end +scheme(f::VarietyFunctionFieldElem) = scheme(parent(f)) +variety(f::VarietyFunctionFieldElem) = variety(parent(f)) + diff --git a/src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl b/src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl index 3ac9c2c3641f..3c59da570bf5 100644 --- a/src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl +++ b/src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl @@ -2329,3 +2329,12 @@ function colength(I::AbsIdealSheaf; covering::Covering=default_covering(scheme(I end return result end + +function is_zero(II::AbsIdealSheaf) + return all(iszero(II(U)) for U in affine_charts(scheme(II))) +end + +function is_zero(II::PrimeIdealSheafFromChart) + return is_zero(II(original_chart(II))) +end + From bf93ffa76e48f9cc228d2840e0b6f5c9110f797d Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 14:35:32 +0100 Subject: [PATCH 26/41] Add tests. --- .../Schemes/self_intersection.jl | 26 +++++++++++++++++++ 1 file changed, 26 insertions(+) create mode 100644 test/AlgebraicGeometry/Schemes/self_intersection.jl diff --git a/test/AlgebraicGeometry/Schemes/self_intersection.jl b/test/AlgebraicGeometry/Schemes/self_intersection.jl new file mode 100644 index 000000000000..7a843e6326bb --- /dev/null +++ b/test/AlgebraicGeometry/Schemes/self_intersection.jl @@ -0,0 +1,26 @@ +@testset "self intersection" begin + IP3 = projective_space(QQ, [:x, :y, :z, :w]) + S = homogeneous_coordinate_ring(IP3) + (x, y, z, w) = gens(S) + # The following line produces an example where the current heuristic + # for moving around divisors to general position still runs into an + # infinite loop. We would like to solve this problem in the long run. + #I = ideal(S, x^3 + y*z*w) + # This example is fine, though. + I = ideal(S, x^2 + y*z) + IPX, inc_IPX = sub(IP3, I) + X = covered_scheme(IPX) + J = ideal(homogeneous_coordinate_ring(IPX), [x, y, z]) + JJ = ideal_sheaf(IPX, J) + bl = blow_up(JJ) + Y = domain(bl) + @test is_smooth(Y) + E = exceptional_divisor(bl) + E_weil = irreducible_decomposition(weil_divisor(E)) + @test integral(Oscar._intersect(E, E_weil)) == -2 + mov = Oscar.move_divisor(E_weil) + mov = Oscar.move_divisor(mov) + #mov = Oscar.move_divisor(Oscar.move_divisor(Oscar.move_divisor(E_weil))) + @test integral(Oscar._intersect(E, mov)) == -2 +end + From 318dbef3e81b2e7780469caa0c159931af8dffd5 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 14:39:30 +0100 Subject: [PATCH 27/41] Disable some tests for the moment. --- experimental/Durham2024/test/runtests.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/experimental/Durham2024/test/runtests.jl b/experimental/Durham2024/test/runtests.jl index 7001b1965b7f..770687414503 100644 --- a/experimental/Durham2024/test/runtests.jl +++ b/experimental/Durham2024/test/runtests.jl @@ -1,3 +1,3 @@ include("fixes.jl") -include("runtests2.jl") +#include("runtests2.jl") # disabled for the moment; needs cleaning. From 4296b5b660bf3f10faf185b4ab8f172cc2b8d8cb Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 14:43:45 +0100 Subject: [PATCH 28/41] Remove Durham file for the moment (to be added in a separate PR). --- experimental/Durham2024/README.md | 12 --- experimental/Durham2024/docs/doc.main | 7 -- experimental/Durham2024/src/Durham2024.jl | 15 --- .../Durham2024/src/erroxefunctions.jl | 8 -- experimental/Durham2024/src/fixes.jl | 11 --- experimental/Durham2024/test/fixes.jl | 38 -------- experimental/Durham2024/test/intersections.jl | 96 ------------------- experimental/Durham2024/test/runtests.jl | 3 - experimental/Durham2024/test/runtests2.jl | 33 ------- 9 files changed, 223 deletions(-) delete mode 100644 experimental/Durham2024/README.md delete mode 100644 experimental/Durham2024/docs/doc.main delete mode 100644 experimental/Durham2024/src/Durham2024.jl delete mode 100644 experimental/Durham2024/src/erroxefunctions.jl delete mode 100644 experimental/Durham2024/src/fixes.jl delete mode 100644 experimental/Durham2024/test/fixes.jl delete mode 100644 experimental/Durham2024/test/intersections.jl delete mode 100644 experimental/Durham2024/test/runtests.jl delete mode 100644 experimental/Durham2024/test/runtests2.jl diff --git a/experimental/Durham2024/README.md b/experimental/Durham2024/README.md deleted file mode 100644 index 564c57c64cc7..000000000000 --- a/experimental/Durham2024/README.md +++ /dev/null @@ -1,12 +0,0 @@ -# Contributions to OSCAR from the Computational Algebraic Geometry workshop in Durham, 2024 - -## Aims - -This folder contains contributions made by participants of the workshop -on Computational Algebraic Geometry, held in Durham, November 18-22nd, 2024. - -## Status - -This is work in progress which will be updated throughout the week and eventually -cleaned up and finalized for a proper pull request to OSCAR. - diff --git a/experimental/Durham2024/docs/doc.main b/experimental/Durham2024/docs/doc.main deleted file mode 100644 index 6bc6bb7bd30c..000000000000 --- a/experimental/Durham2024/docs/doc.main +++ /dev/null @@ -1,7 +0,0 @@ -# Additional functionality for intersection theory on schemes - -## Pushforward on algebraic cycles along proper morphisms -```@docs - pushforward(p::BlowupMorphism, W::AbsAlgebraicCycle) -``` - diff --git a/experimental/Durham2024/src/Durham2024.jl b/experimental/Durham2024/src/Durham2024.jl deleted file mode 100644 index 00e6b16f5372..000000000000 --- a/experimental/Durham2024/src/Durham2024.jl +++ /dev/null @@ -1,15 +0,0 @@ -# pushforward of algebraic cycles for proper maps - -@doc raw""" - pushforward(p::BlowupMorphism, W::AbsAlgebraicCycle) - -Given a `BlowupMorphism` ``p : X → Y`` and an `AbsAlgebraicCycle` ``W ∈ Z_k(X)``, -compute ``p_*(W) ∈ Z_k(Y)``. -""" -function pushforward(p::BlowupMorphism, W::AbsAlgebraicCycle) - error("not implemented") -end - -include("fixes.jl") -include("erroxefunctions.jl") - diff --git a/experimental/Durham2024/src/erroxefunctions.jl b/experimental/Durham2024/src/erroxefunctions.jl deleted file mode 100644 index c35f4c9ec980..000000000000 --- a/experimental/Durham2024/src/erroxefunctions.jl +++ /dev/null @@ -1,8 +0,0 @@ -function embedding(C::EffectiveCartierDivisor) - I=ideal_sheaf(C); - C1,inc_C = sub(I); - return inc_C -end - -function reduce_to_surface() -end \ No newline at end of file diff --git a/experimental/Durham2024/src/fixes.jl b/experimental/Durham2024/src/fixes.jl deleted file mode 100644 index d0360c1b7a27..000000000000 --- a/experimental/Durham2024/src/fixes.jl +++ /dev/null @@ -1,11 +0,0 @@ -function canonical_divisor(X:: AbsAffineScheme{<:Field, <:MPolyRing}) - return CartierDivisor(covered_scheme(X), ZZ) -end - -function self_intersection_via_adjunction(K_X::CartierDivisor, C::EffectiveCartierDivisor, g::Int) - return 2g-2-intersect(K_X, C) -end -function self_intersection_via_adjunction(K_X::CartierDivisor, C::CartierDivisor, g::Int) - return 2g-2-sum(a*intersect(K_X, comp) for (a, comp) in coefficient_dict(C); init=0) -end - diff --git a/experimental/Durham2024/test/fixes.jl b/experimental/Durham2024/test/fixes.jl deleted file mode 100644 index bcfcf1d38433..000000000000 --- a/experimental/Durham2024/test/fixes.jl +++ /dev/null @@ -1,38 +0,0 @@ -@testset "components of cycles" begin - IA2 = affine_space(QQ, [:x, :y]) - bl0 = blow_up(IA2, ideal(OO(IA2), gens(OO(IA2)))) - E = exceptional_divisor(bl0) - EW = weil_divisor(E) - c = first(components(WeilDivisor, EW)) - D = Oscar.underlying_cycle(EW) - cc = first(components(AlgebraicCycle, D)) - @test EW[c] == D[cc] - - X = domain(bl0) - FX = function_field(X) - U = first(affine_charts(X)) - x = first(gens(OO(U))) - f = FX(x) - D = weil_divisor(f) - @test 2*D == weil_divisor(f^2) - - (s, x, y) = ambient_coordinates(U) - I = ideal_sheaf(E) - IU = I(U) - @test OO(U)(x) in IU - @test OO(U)(y) in IU - @test !(OO(U)(s) in IU) - - f = FX(x) - W = weil_divisor(f) - I1, I2 = components(W) - @test I == I1 || I == I2 - @test I != I1 || I != I2 - - D = weil_divisor(E) - D2 = irreducible_decomposition(D - W) - @test integral(intersect(D2, E)) == -1 - - @test integral(intersect(Oscar.move_divisor(D), E)) == -1 -end - diff --git a/experimental/Durham2024/test/intersections.jl b/experimental/Durham2024/test/intersections.jl deleted file mode 100644 index c9943944fb25..000000000000 --- a/experimental/Durham2024/test/intersections.jl +++ /dev/null @@ -1,96 +0,0 @@ -P3 = projective_space(QQ,[:x0,:x1,:x2,:x3]); -(x0,x1,x2,x3) = homogeneous_coordinates(P3); -R=homogeneous_coordinate_ring(P3); -######################################## -I = ideal(R,[x0,x3]); -#L,inc_L = sub(P3,I); -I1 = ideal(R,[x0-x1,x2]); -#L1,inc_L1 = sub(P3, I1); -I2 = ideal(R,[x1,x2]); -#L2,inc_L2 = sub(P3,ideal(R,[x1,x2])); -I3 = ideal(R,[x1,x2-x3]); -#L3,inc_L3 = sub(P3,ideal(R,[x1,x2-x3])); -# (X2,E2,inc_E2) = blowup(inc_L2); This does not work. -IH = ideal(x0-x2); -########################################## -Y = covered_scheme(P3); -II1=ideal_sheaf(P3,I1); -II2=ideal_sheaf(P3,I2); -II3=ideal_sheaf(P3,I3); -II=ideal_sheaf(P3,I); -IIH=ideal_sheaf(P3,IH); -H=effective_cartier_divisor(IIH); -############################################### -#We do the first blowup, by blowing up L -bl = blow_up(Y,II); -X= domain(bl); -II11=strict_transform(bl,II1); -II12=strict_transform(bl,II2); -II13=strict_transform(bl,II3); -H1=strict_transform(bl,H); -E=exceptional_divisor(bl); -################################################## -#We do the second blowup, by blowing up L2 -bl2 = blow_up(X,II12); -X2= domain(bl2); -II21=strict_transform(bl2,II11); -II23=strict_transform(bl2,II13); -H2=strict_transform(bl2,H1); -EE=strict_transform(bl2,E); -E2=exceptional_divisor(bl2); -################################################### -#We do the third blowup, by blowing up L1 -bl1 = blow_up(X2,II21); -X1= domain(bl1); -II33=strict_transform(bl1,II23); -H3=strict_transform(bl1,H2); -EEE=strict_transform(bl1,EE); -EE2=strict_transform(bl1,E2); -E1=exceptional_divisor(bl1); -#################################################### -#We do the fourth blowup, by blowing up L3 -bl3 = blow_up(X1,II33); -X_final= domain(bl3); -H_final = strict_transform(bl3,H3); -E_final=strict_transform(bl3,EEE); -E2_final=strict_transform(bl3,EE2); -E1_final=strict_transform(bl3,E1); -E3_final=exceptional_divisor(bl3); -################################################### -Int1=intersect(E_final,E1_final); -Int23=intersect(E2_final,E3_final); -Int21=intersect(E2_final,E1_final); -E2E3E1=intersect(Int23,E1_final); -integral(E2E3E1); -################################################## -#Now we want to do intersections inside the surface. Lets start with E2. -inc_E2 = embedding(E2_final); -l21=pullback(inc_E2,E1_final); -l23=pullback(inc_E2,E3_final); -Int123=intersect(l21,l23); -integral(Int123); -##################################################### -inc_E1 = embedding(E1_final); -l21=pullback(inc_E2,E1_final); -l23=pullback(inc_E2,E3_final); -Int123=intersect(l21,l23); -integral(Int123); -kX=-4*H_final+E1_final+E2_final+E3_final+E_final; -################################################### -inc_H = embedding(H_final); -l21=pullback(inc_E2,E1_final); -l23=pullback(inc_E2,E3_final); -Int123=intersect(l21,l23); -integral(Int123); - -E2_sub, inc_E2 = sub(ideal_sheaf(E2_final)) -pb_E1 = pullback(inc_E2, E1_final) -pb_E1_weil = weil_divisor(pb_E1) -int1 = integral(Oscar._intersect(pb_E1, pb_E1_weil)) -move = Oscar.move_divisor(pb_E1_weil) -int2 = integral(Oscar._intersect(pb_E1, move)) -@test int1 == int2 - - - - diff --git a/experimental/Durham2024/test/runtests.jl b/experimental/Durham2024/test/runtests.jl deleted file mode 100644 index 770687414503..000000000000 --- a/experimental/Durham2024/test/runtests.jl +++ /dev/null @@ -1,3 +0,0 @@ -include("fixes.jl") -#include("runtests2.jl") # disabled for the moment; needs cleaning. - diff --git a/experimental/Durham2024/test/runtests2.jl b/experimental/Durham2024/test/runtests2.jl deleted file mode 100644 index 1980ada89f85..000000000000 --- a/experimental/Durham2024/test/runtests2.jl +++ /dev/null @@ -1,33 +0,0 @@ -@testset "canonical divisor of a Kleinian singularity" begin -# We calculate the canonical divisor of the blow up of a Kleinian singularity in this document. -R, (x, y, z) = QQ[:x, :y, :z] -I = ideal(R, [x^3 - y*z]) - -IA3 = spec(R) -X, inc_X = sub(IA3, I) #X is the A2 Kleinian singularity -I_sing = ideal(R, gens(R)) - -pr1 = blow_up(IA3, I_sing) #blow up the affine space IA3 -bl_IA3 = domain(pr1) -E = exceptional_divisor(pr1) -bl_X, inc_bl_X, pr2 = strict_transform(pr1, inc_X) #bl_X is the blow up of X - -EX = pullback(inc_bl_X, E) -EX_weil = weil_divisor(EX) -EX_weil_decom = irreducible_decomposition(EX_weil) -C1 = components(WeilDivisor, EX_weil_decom)[1] -C2 = components(WeilDivisor, EX_weil_decom)[2] - -bl_X_canonical_divisor = weil_divisor(bl_X, ZZ) -Z11 = Oscar.self_intersection_via_adjunction(bl_X_canonical_divisor, C1, 0) -Z22 = Oscar.self_intersection_via_adjunction(bl_X_canonical_divisor, C2, 0) -Z12 = intersect(C1, C2) -Z21 = intersect(C2, C1) -Intersection_matrix = ZZ[Z11 Z12; Z21 Z22] -end - - - - - - From 84ba621cb749d1386c99f4db2938d7f3078231f2 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 14:44:23 +0100 Subject: [PATCH 29/41] Clean up inclusion order. --- experimental/Experimental.jl | 1 - 1 file changed, 1 deletion(-) diff --git a/experimental/Experimental.jl b/experimental/Experimental.jl index 6b418b620d83..878a6e356922 100644 --- a/experimental/Experimental.jl +++ b/experimental/Experimental.jl @@ -13,7 +13,6 @@ const orderedpkgs = [ "Schemes", "FTheoryTools", # must be loaded after Schemes and LieAlgebras "IntersectionTheory", # must be loaded after Schemes - "Durham2024", ] const exppkgs = filter(x->isdir(joinpath(expdir, x)) && !(x in orderedpkgs), readdir(expdir)) append!(exppkgs, orderedpkgs) From 007694cd40c06931eb8f0dd502f30898710c6fe1 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 15:35:37 +0100 Subject: [PATCH 30/41] Some cleaning up. --- .../Schemes/Divisors/AlgebraicCycles.jl | 12 ++++++------ .../Schemes/Divisors/WeilDivisor.jl | 6 ++---- test/AlgebraicGeometry/Schemes/self_intersection.jl | 1 + 3 files changed, 9 insertions(+), 10 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl index e1f4d9126184..fe1c31984119 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl @@ -461,11 +461,11 @@ function irreducible_decomposition(D::AbsAlgebraicCycle) for P in keys(coefficient_dict(D)) found = false for bucket in buckets - if P == first(bucket) - push!(bucket, P) - found = true - break - end + if P == first(bucket) + push!(bucket, P) + found = true + break + end end !found && push!(buckets, [P]) end @@ -476,7 +476,7 @@ function irreducible_decomposition(D::AbsAlgebraicCycle) is_zero(c) && continue coeff_dict[first(bucket)] = c end - return AlgebraicCycle(scheme(D), coefficient_ring(D), coeff_dict; check=false) + return AlgebraicCycle(ambient_scheme(D), coefficient_ring(D), coeff_dict; check=false) end @vprint :Divisors 4 " decomposing components" result = zero(D) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index 0ba27e139e80..10bc5639fe00 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -652,8 +652,6 @@ function _colength_in_localization(I::Ideal, P::Ideal) y = y*x k += 1 end - - return _colength_in_localization(saturated_ideal(I), saturated_ideal(P)) end # same assumptions as above apply. @@ -780,14 +778,14 @@ function move_divisor( result = irreducible_decomposition(D - weil_divisor(f)) # Check whether the supports are really different if any(any(P == Q for Q in components(D)) for P in components(result)) - @show "switched component" return move_divisor(D; randomization=true, check, is_prime) end return result end function is_zero(D::AbsAlgebraicCycle) - return all(is_zero(c) || is_one(I) for (I, c) in coefficient_dict(D)) + all(is_zero(c) || is_one(I) for (I, c) in coefficient_dict(D)) && return true + return all(is_zero(c) || is_one(I) for (I, c) in coefficient_dict(irreducible_decomposition(D))) && return true end # Internal method which performs an automated move of the second argument diff --git a/test/AlgebraicGeometry/Schemes/self_intersection.jl b/test/AlgebraicGeometry/Schemes/self_intersection.jl index 7a843e6326bb..af50152a8972 100644 --- a/test/AlgebraicGeometry/Schemes/self_intersection.jl +++ b/test/AlgebraicGeometry/Schemes/self_intersection.jl @@ -20,6 +20,7 @@ @test integral(Oscar._intersect(E, E_weil)) == -2 mov = Oscar.move_divisor(E_weil) mov = Oscar.move_divisor(mov) + # The following line will also produce an infinite loop. TODO: Fix this! #mov = Oscar.move_divisor(Oscar.move_divisor(Oscar.move_divisor(E_weil))) @test integral(Oscar._intersect(E, mov)) == -2 end From d12182f1d7a07358b86ad178805122b26d1ddfcf Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 20:00:33 +0100 Subject: [PATCH 31/41] Address comments from review. --- .../Schemes/Divisors/AlgebraicCycles.jl | 61 +++++++++++-------- .../Schemes/Divisors/WeilDivisor.jl | 7 ++- 2 files changed, 40 insertions(+), 28 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl index fe1c31984119..84cf9e4b7131 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl @@ -455,32 +455,12 @@ where the `components` ``Iₖ`` of ``E`` are all sheaves of prime ideals. """ function irreducible_decomposition(D::AbsAlgebraicCycle) @vprint :Divisors 4 "computing irreducible decomposition for $D" - if all(is_prime, keys(coefficient_dict(D))) - @vprint :Divisors 4 " all components are already prime, cleaning up..." - buckets = Vector{Vector{AbsIdealSheaf}}() - for P in keys(coefficient_dict(D)) - found = false - for bucket in buckets - if P == first(bucket) - push!(bucket, P) - found = true - break - end - end - !found && push!(buckets, [P]) - end - R = coefficient_ring(D) - coeff_dict = IdDict{AbsIdealSheaf, elem_type(R)}() - for bucket in buckets - c = sum(D[P] for P in bucket; init=zero(R)) - is_zero(c) && continue - coeff_dict[first(bucket)] = c - end - return AlgebraicCycle(ambient_scheme(D), coefficient_ring(D), coeff_dict; check=false) - end - @vprint :Divisors 4 " decomposing components" result = zero(D) for (I, a) in coefficient_dict(D) + if is_prime(I) + result[I] = a + continue + end next_dict = IdDict{AbsIdealSheaf, elem_type(coefficient_ring(D))}() decomp = maximal_associated_points(I) for P in decomp @@ -489,7 +469,34 @@ function irreducible_decomposition(D::AbsAlgebraicCycle) end result = result + a * AlgebraicCycle(ambient_scheme(D), coefficient_ring(D), next_dict, check=false) end - return result + return _unique_prime_components(result) +end + +# Given a cycle `D` with only prime components, compare the components +# and gather the coefficients of equal ones so that the result has +# pairwise distinct components. +function _unique_prime_components(D::AbsAlgebraicCycle) + @hassert :Divisors 2 all(is_prime, keys(coefficient_dict(D))) + buckets = Vector{Vector{AbsIdealSheaf}}() + for P in keys(coefficient_dict(D)) + found = false + for bucket in buckets + if P == first(bucket) + push!(bucket, P) + found = true + break + end + end + !found && push!(buckets, [P]) + end + R = coefficient_ring(D) + coeff_dict = IdDict{AbsIdealSheaf, elem_type(R)}() + for bucket in buckets + c = sum(D[P] for P in bucket; init=zero(R)) + is_zero(c) && continue + coeff_dict[first(bucket)] = c + end + return AlgebraicCycle(ambient_scheme(D), coefficient_ring(D), coeff_dict; check=false) end function _colength_in_localization(Q::AbsIdealSheaf, P::AbsIdealSheaf; covering=simplified_covering(scheme(P))) @@ -562,13 +569,13 @@ end function components(::Type{T}, D::AbsAlgebraicCycle) where {T <: AbsAlgebraicCycle} X = scheme(D) R = coefficient_ring(D) - return [AlgebraicCycle(X, R, IdDict{AbsIdealSheaf, elem_type(R)}([I=>one(R)]); check=false) for I in components(D)] + return [AlgebraicCycle(X, R, IdDict{AbsIdealSheaf, elem_type(R)}([I=>one(R)]); check=false)::T for I in components(D)] end function components(::Type{T}, D::AbsWeilDivisor) where {T <: AbsWeilDivisor} X = scheme(D) R = coefficient_ring(D) - return [WeilDivisor(X, R, IdDict{AbsIdealSheaf, elem_type(R)}([I=>one(R)]); check=false) for I in components(D)] + return [WeilDivisor(X, R, IdDict{AbsIdealSheaf, elem_type(R)}([I=>one(R)]); check=false)::T for I in components(D)] end function getindex(D::AbsAlgebraicCycle, C::AbsAlgebraicCycle) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index 10bc5639fe00..3fc5275ab9e1 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -675,6 +675,11 @@ function weil_divisor( X = scheme(f) ideal_dict = IdDict{AbsIdealSheaf, elem_type(ring)}() for U in patches(covering) + # TODO: We compute the dimensions again and again. + # Instead we should store these things in some matured version of `decomposition_info`! + has_decomposition_info(covering) && dim(ideal(OO(U), decomposition_info(covering)[U])) <= dim(X) - 2 && continue + + # covering to take a shortcut in the @vprint :Divisors 4 "doing patch $U\n" inc_dict = IdDict{AbsIdealSheaf, elem_type(ring)}() f_loc = f[U] @@ -729,7 +734,7 @@ end Given an `AbsWeilDivisor` `D` on a scheme `X`, create a principal divisor `div(f)` for some rational function, so that `D - div(f)` is supported -away from the original support of `D`. +in (non-closed) scheme-theoretic points which are different from those of `D`. Keyword arguments: * `randomization`: By default, the choices made to create `f` keep it as simple as possible. However, one might encounter constellations where this will just swap two components of `D`. In order to avoid this, one can then switch on randomization here. From d55d8935406381d95a2ab0442c2550bb61343749 Mon Sep 17 00:00:00 2001 From: Matthias Zach <85350711+HechtiDerLachs@users.noreply.github.com> Date: Mon, 25 Nov 2024 20:01:15 +0100 Subject: [PATCH 32/41] Update src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl Co-authored-by: Simon Brandhorst --- src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index 3fc5275ab9e1..45e1ba9b6cb5 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -667,6 +667,10 @@ function _find_principal_generator(I::Union{<:MPolyLocalizedIdeal, <:MPolyQuoLoc end # produce the principal divisor associated to a rational function +function principal_divisor(::Type{WeilDivisor}, f::VarietyFunctionFieldElem; + ring::Ring=ZZ, covering::Covering=default_covering(scheme(f)) + ) = weil_divisor(f; ring; covering) + function weil_divisor( f::VarietyFunctionFieldElem; ring::Ring=ZZ, covering::Covering=default_covering(scheme(f)) From 7c61fa5612708fbdb3aee47e21a32deb102c1370 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Mon, 25 Nov 2024 20:44:34 +0100 Subject: [PATCH 33/41] Attempt to fix up code. --- src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index 45e1ba9b6cb5..6b01c3b73266 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -681,7 +681,7 @@ function weil_divisor( for U in patches(covering) # TODO: We compute the dimensions again and again. # Instead we should store these things in some matured version of `decomposition_info`! - has_decomposition_info(covering) && dim(ideal(OO(U), decomposition_info(covering)[U])) <= dim(X) - 2 && continue + has_decomposition_info(covering) && (dim(ideal(OO(U), decomposition_info(covering)[U])) <= dim(X) - 2) && continue # covering to take a shortcut in the @vprint :Divisors 4 "doing patch $U\n" From 17048b366e82f7e766970310b04c47c44793810e Mon Sep 17 00:00:00 2001 From: Matthias Zach <85350711+HechtiDerLachs@users.noreply.github.com> Date: Mon, 25 Nov 2024 20:54:59 +0100 Subject: [PATCH 34/41] Update src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl --- src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index 6b01c3b73266..b8b384f42767 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -667,9 +667,9 @@ function _find_principal_generator(I::Union{<:MPolyLocalizedIdeal, <:MPolyQuoLoc end # produce the principal divisor associated to a rational function -function principal_divisor(::Type{WeilDivisor}, f::VarietyFunctionFieldElem; +principal_divisor(::Type{WeilDivisor}, f::VarietyFunctionFieldElem; ring::Ring=ZZ, covering::Covering=default_covering(scheme(f)) - ) = weil_divisor(f; ring; covering) + ) = weil_divisor(f; ring, covering) function weil_divisor( f::VarietyFunctionFieldElem; From 4a6f57c6725f28d76aecdffb098594042b6ae12e Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Tue, 26 Nov 2024 09:47:29 +0100 Subject: [PATCH 35/41] scheme -> ambient_scheme. --- src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index b8b384f42767..7d4226d53faa 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -750,7 +750,7 @@ function move_divisor( randomization::Bool=false, is_prime::Bool=false ) - X = scheme(D) + X = ambient_scheme(D) @check is_irreducible(X) && is_reduced(X) "scheme must be irreducible and reduced" is_zero(D) && return D @@ -803,8 +803,8 @@ end # its base field. But the output is only valid if that is true. # Therefore, we have no choice, but to hide it from the user for now. function _intersect(C::CartierDivisor, D::AbsWeilDivisor) - X = scheme(C) - @assert X === scheme(D) + X = ambient_scheme(C) + @assert X === ambient_scheme(D) R = coefficient_ring(C) @assert R === coefficient_ring(D) result = AlgebraicCycle(X, R) @@ -815,8 +815,8 @@ function _intersect(C::CartierDivisor, D::AbsWeilDivisor) end function _intersect(E::EffectiveCartierDivisor, D::AbsWeilDivisor; check::Bool=true) - X = scheme(E) - @assert X === scheme(D) + X = ambient_scheme(E) + @assert X === ambient_scheme(D) R = coefficient_ring(D) result = AlgebraicCycle(X, R) cpcd = copy(coefficient_dict(D)) From 2a65bacc2d453fc6e2c513ab14ffceb55a605bcd Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Tue, 26 Nov 2024 10:04:55 +0100 Subject: [PATCH 36/41] Address concerns from review. --- .../Schemes/Divisors/AlgebraicCycles.jl | 2 +- .../Schemes/Divisors/WeilDivisor.jl | 12 +++++++++--- 2 files changed, 10 insertions(+), 4 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl index 84cf9e4b7131..6058acc6b333 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl @@ -451,7 +451,7 @@ end irreducible_decomposition(D::AbsAlgebraicCycle) Return a cycle ``E`` equal to ``D`` but as a formal sum ``E = ∑ₖ aₖ ⋅ Iₖ`` -where the `components` ``Iₖ`` of ``E`` are all sheaves of prime ideals. +where the `components` ``Iₖ`` of ``E`` are pairwise distinct sheaves of prime ideals. """ function irreducible_decomposition(D::AbsAlgebraicCycle) @vprint :Divisors 4 "computing irreducible decomposition for $D" diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index 7d4226d53faa..caae7144a936 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -736,9 +736,15 @@ end @doc raw""" move_divisor(D::AbsWeilDivisor; check::Bool=false) -Given an `AbsWeilDivisor` `D` on a scheme `X`, create a principal divisor -`div(f)` for some rational function, so that `D - div(f)` is supported -in (non-closed) scheme-theoretic points which are different from those of `D`. +Given an `AbsWeilDivisor` `D` on a scheme `X`, apply a heuristic attempt +to create a principal divisor `div(f)` for some rational function, +so that the support of the divisor `D - div(f)` does not contain the +generic scheme-theoretic points in the support of `D`. + +Note that `supp(D - div(f)) ∩ supp(D)` need not be empty! The point is that +the minimal associated primes of the support of `D - div(f)` should be different +from the minimal associated primes of `D`. This is experimental and might not +always succeed. Keyword arguments: * `randomization`: By default, the choices made to create `f` keep it as simple as possible. However, one might encounter constellations where this will just swap two components of `D`. In order to avoid this, one can then switch on randomization here. From a1abedda8a256bfcf620b3eacb27193d5abce214 Mon Sep 17 00:00:00 2001 From: Matthias Zach <85350711+HechtiDerLachs@users.noreply.github.com> Date: Wed, 27 Nov 2024 17:03:52 +0100 Subject: [PATCH 37/41] Clarify the docu for `move_divisor`. --- src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index caae7144a936..682c0d10493c 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -738,8 +738,7 @@ end Given an `AbsWeilDivisor` `D` on a scheme `X`, apply a heuristic attempt to create a principal divisor `div(f)` for some rational function, -so that the support of the divisor `D - div(f)` does not contain the -generic scheme-theoretic points in the support of `D`. +so that `supp(D - div(f)) ∩ supp(D)` has codimension greater or equal to `2`. In other words: We try to move `D` away from its original position as much as possible within its rational equivalence class. Note that `supp(D - div(f)) ∩ supp(D)` need not be empty! The point is that the minimal associated primes of the support of `D - div(f)` should be different From 5256ece7186ad0e6e94e09e66f4198e56024c973 Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Thu, 28 Nov 2024 12:20:26 +0100 Subject: [PATCH 38/41] More use of decomposition info. --- .../Schemes/Divisors/WeilDivisor.jl | 35 +++++++++++++++---- 1 file changed, 28 insertions(+), 7 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index 682c0d10493c..f8b90f58a744 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -681,7 +681,9 @@ function weil_divisor( for U in patches(covering) # TODO: We compute the dimensions again and again. # Instead we should store these things in some matured version of `decomposition_info`! - has_decomposition_info(covering) && (dim(ideal(OO(U), decomposition_info(covering)[U])) <= dim(X) - 2) && continue + has_dec_inf = has_decomposition_info(covering) + dec_inf_id = has_dec_inf ? ideal(OO(U), decomposition_info(covering)[U]) : ideal(OO(U), elem_type(OO(U))[]) + has_dec_inf && (dim(dec_inf_id) <= dim(X) - 2) && continue # covering to take a shortcut in the @vprint :Divisors 4 "doing patch $U\n" @@ -689,27 +691,46 @@ function weil_divisor( f_loc = f[U] num = numerator(f_loc) den = denominator(f_loc) + + # If present, we can exploit the decomposition info here, too. + # The point is the following: In a new chart, we only need to + # deal with scheme theoretic points (of codimension 1) which + # are not visible in the other charts. Algebraically this means + # that we can add the local generators of the decomposition + # info to the numerator- and denominator ideals and only work + # with the minimal associated primes (of codimension one) of + # those ideals. num_ideal = ideal(OO(U), num) den_ideal = ideal(OO(U), den) - num_dec = primary_decomposition(num_ideal) - den_dec = primary_decomposition(den_ideal) + + if has_dec_inf + num_ideal = num_ideal + dec_inf_id + den_ideal = den_ideal + dec_inf_id + end + + num_dec = minimal_primes(num_ideal) + den_dec = minimal_primes(den_ideal) @vprint :Divisors 4 " numerator:\n" - for (_, P) in num_dec + for P in num_dec + # In case of use of decomposition info, we need to discard primes of + has_dec_inf && (dim(P) < dim(X) - 1) && continue @vprint :Divisors 4 " $P\n" # If this component was already seen in another patch, skip it. new_comp = PrimeIdealSheafFromChart(X, U, P) @vprint :Divisors 4 " $(any(new_comp == PP for PP in keys(ideal_dict)) ? "already found" : "new component")\n" - any(new_comp == PP for PP in keys(ideal_dict)) && continue + !has_dec_inf && any(new_comp == PP for PP in keys(ideal_dict)) && continue c = _colength_in_localization(num_ideal, P) @vprint :Divisors 4 " multiplicity $c\n" inc_dict[new_comp] = c end @vprint :Divisors 4 " denominator:\n" - for (_, P) in den_dec + for P in den_dec + # In case of use of decomposition info, we need to discard primes of + has_dec_inf && (dim(P) < dim(X) - 1) && continue # If this component was already seen in another patch, skip it. new_comp = PrimeIdealSheafFromChart(X, U, P) @vprint :Divisors 4 " $(any(new_comp == PP for PP in keys(ideal_dict)) ? "already found" : "new component")\n" - any(new_comp == PP for PP in keys(ideal_dict)) && continue + !has_dec_inf && any(new_comp == PP for PP in keys(ideal_dict)) && continue c = _colength_in_localization(den_ideal, P) @vprint :Divisors 4 " multiplicity $c\n" key_list = collect(keys(inc_dict)) From 6e5de3f6482b4e945c424ca6b195959994de5ece Mon Sep 17 00:00:00 2001 From: HechtiDerLachs Date: Thu, 28 Nov 2024 13:09:10 +0100 Subject: [PATCH 39/41] Final overhaul of move_divisor. --- .../Schemes/Divisors/WeilDivisor.jl | 35 +++++++++++-------- 1 file changed, 21 insertions(+), 14 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index f8b90f58a744..603c36acc01f 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -774,15 +774,17 @@ function move_divisor( D::AbsWeilDivisor; check::Bool=false, randomization::Bool=false, - is_prime::Bool=false + is_prime::Bool=false, + rec_depth::Int=0 ) + rec_depth > 5 && error("the current constellation seems to lead to infinite loops for the current implementation") X = ambient_scheme(D) @check is_irreducible(X) && is_reduced(X) "scheme must be irreducible and reduced" is_zero(D) && return D if !is_prime && !Oscar.is_prime(D) R = coefficient_ring(D) - return sum(a*move_divisor(WeilDivisor(D, R; check=false)) for (D, a) in coefficient_dict(irreducible_decomposition(D)); init=WeilDivisor(X, R)) + return sum(a*move_divisor(WeilDivisor(D, R; check=false); rec_depth) for (D, a) in coefficient_dict(irreducible_decomposition(D)); init=WeilDivisor(X, R)) end # We may assume that `D` is prime. @@ -801,19 +803,28 @@ function move_divisor( # principal there. g = gens(saturated_ideal(I)) g = sort!(g, by=total_degree) - i = findfirst(f->(ideal(L, f) == LP), g) - x = g[i] + x = zero(base_ring(I)) + kk = base_ring(X) + RP = ambient_coordinate_ring(U) + if randomization + x = sum(rand(kk, 1:10)*x for x in g; init=zero(RP)) + while !(ideal(L, x) == LP) + # try again. + x = sum(rand(kk, 1:10)*x for x in g; init=zero(RP)) + end + else + i = findfirst(f->(ideal(L, f) == LP), g) + x = g[i] + end f = function_field(X; check)(x) if randomization - kk = base_ring(X) - R = ambient_coordinate_ring(U) - y = rand(kk, 1:10) + sum(rand(kk, 1:10)*a for a in gens(R); init=zero(R)) + y = rand(kk, 1:10) + sum(rand(kk, 1:10)*a for a in gens(RP); init=zero(RP)) f = f*inv(parent(f)(y)) end result = irreducible_decomposition(D - weil_divisor(f)) # Check whether the supports are really different if any(any(P == Q for Q in components(D)) for P in components(result)) - return move_divisor(D; randomization=true, check, is_prime) + return move_divisor(D; randomization=true, check, is_prime, rec_depth=rec_depth+1) end return result end @@ -854,14 +865,10 @@ function _intersect(E::EffectiveCartierDivisor, D::AbsWeilDivisor; check::Bool=t # We have to think about how this can effectively be avoided. See the test file # for an example. if is_zero(pullback(inc_P, ideal_sheaf(E))) - P_moved = move_divisor(WeilDivisor(X, R, - IdDict{AbsIdealSheaf, elem_type(R)}( - [P=>one(R)]); check=false); check=false, randomization=true, is_prime=true) + P_moved = move_divisor(weil_divisor(P, R; check=false); randomization=false, is_prime=true) result = result + a*_intersect(E, P_moved) else - result = result + a*AlgebraicCycle(X, R, - IdDict{AbsIdealSheaf, elem_type(R)}( - [P + ideal_sheaf(E)=>one(R)]); check=false) + result = result + a*algebraic_cycle(P + ideal_sheaf(E), R; check=false) end end return result From 3aaac1259242184dd46c87495dfdcf7baeb25079 Mon Sep 17 00:00:00 2001 From: Matthias Zach <85350711+HechtiDerLachs@users.noreply.github.com> Date: Thu, 28 Nov 2024 13:11:51 +0100 Subject: [PATCH 40/41] Update src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl Co-authored-by: Simon Brandhorst --- src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl index 6058acc6b333..9562efc9b70d 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl @@ -580,7 +580,7 @@ end function getindex(D::AbsAlgebraicCycle, C::AbsAlgebraicCycle) comps = components(C) - @assert isone(length(comps)) + @req isone(length(comps)) "$(C) must consist of a single component only" return D[first(comps)] end From 9a25b20f5cbcad83d916549fd267372c49fa1de1 Mon Sep 17 00:00:00 2001 From: Simon Brandhorst Date: Thu, 28 Nov 2024 15:26:50 +0100 Subject: [PATCH 41/41] Update src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl --- src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl | 9 ++------- 1 file changed, 2 insertions(+), 7 deletions(-) diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index 603c36acc01f..1f0138973197 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -703,13 +703,8 @@ function weil_divisor( num_ideal = ideal(OO(U), num) den_ideal = ideal(OO(U), den) - if has_dec_inf - num_ideal = num_ideal + dec_inf_id - den_ideal = den_ideal + dec_inf_id - end - - num_dec = minimal_primes(num_ideal) - den_dec = minimal_primes(den_ideal) + num_dec = minimal_primes(num_ideal + dec_inf_id) + den_dec = minimal_primes(den_ideal + dec_inf_id) @vprint :Divisors 4 " numerator:\n" for P in num_dec # In case of use of decomposition info, we need to discard primes of