Varieties
Basic invariants
IntersectionTheory.degree — Functiondegree(X::AbsVariety)Compute the degree of $X$ with respect to its polarization.
IntersectionTheory.euler — Functioneuler(X::AbsVariety)
euler(X::TnVariety)Compute the Euler number of a variety $X$.
IntersectionTheory.hilbert_polynomial — Functionhilbert_polynomial(F::AbsBundle)
hilbert_polynomial(X::AbsVariety)Compute the Hilbert polynomial of a bundle $F$ or the Hilbert polynomial of $X$ itself, with respect to the polarization $\mathcal O_X(1)$ on $X$.
IntersectionTheory.milnor — Functionmilnor(X::Variety)Compute the Milnor number $\int_X\mathrm{ch}_n(T_X)$ of a variety $X$.
Examples
julia> degree(grassmannian(2, 4))
2
julia> euler(grassmannian(2, 4))
6
julia> hilbert_polynomial(proj(3))
1//6*t^3 + t^2 + 11//6*t + 1Characteristic classes
By common abuse of notations, the Chern/Todd/... class of a variety will mean the Chern/Todd/... class of its tangent bundle.
IntersectionTheory.chern — Methodchern(X::AbsVariety)
chern(X::TnVariety)Compute the total Chern class of the tangent bundle of $X$.
IntersectionTheory.chern — Methodchern(k::Int, X::AbsVariety)
chern(k::Int, X::TnVariety)Compute the $k$-th Chern class of the tangent bundle of $X$.
IntersectionTheory.chern_number — Functionchern_number(X::Variety, λ::Int...)
chern_number(X::Variety, λ::AbstractVector)Compute the Chern number $c_\lambda (X):=\int_X c_{\lambda_1}(X)\cdots c_{\lambda_k}(X)$, where $\lambda:=(\lambda_1,\dots,\lambda_k)$ is a partition of the dimension of $X$.
IntersectionTheory.chern_numbers — Methodchern_numbers(X::Variety)
chern_numbers(X::Variety, P::Vector{<:Partition})Compute all the Chern numbers of $X$ as a dictionary of $\lambda\Rightarrow c_\lambda(X)$, or only those corresponding to partitions in a given vector.
IntersectionTheory.todd — Methodtodd(X::AbsVariety)
todd(X::TnVariety)Compute the Todd class of the tangent bundle of $X$.
IntersectionTheory.pontryagin — Methodpontryagin(X::AbsVariety)Compute the total Pontryagin class of the tangent bundle of $X$.
IntersectionTheory.pontryagin — Methodpontryagin(k::Int, X::AbsVariety)Compute the $k$-th Pontryagin class of the tangent bundle of $X$.
IntersectionTheory.a_hat_genus — Methoda_hat_genus(X::AbsVariety)Compute the $\hat A$ genus of a variety $X$.
IntersectionTheory.a_hat_genus — Methoda_hat_genus(k::Int, X::AbsVariety)Compute the $k$-th $\hat A$ genus of a variety $X$.
IntersectionTheory.l_genus — Methodl_genus(k::Int, X::AbsVariety)Compute the $k$-th L genus of a variety $X$.
IntersectionTheory.signature — Functionsignature(X::AbsVariety)Compute the signature of a variety $X$.
We also have the following functions for producing generic formulae.
IntersectionTheory.todd — Methodtodd(n::Int)Compute the total class of the Todd genus up to degree $n$, in terms of the Chern classes.
IntersectionTheory.a_hat_genus — Methoda_hat_genus(n::Int)Compute the total class of the $\hat A$ genus up to degree $n$, in terms of the Pontryagin classes.
IntersectionTheory.l_genus — Methodl_genus(n::Int)Compute the total class of the L genus up to degree $n$, in terms of the Pontryagin classes.
Examples
julia> chern(proj(2))
1 + 3*h + 3*h^2
julia> todd(proj(2))
1 + 3//2*h + h^2
julia> pontryagin(proj(2))
1 + 3*h^2
julia> chern_number(proj(3), [2,1])
24
julia> chern_numbers(proj(3))
Dict{AbstractAlgebra.Generic.Partition{Int64}, Nemo.QQFieldElem} with 3 entries:
2₁1₁ => 24
3₁ => 4
1₃ => 64
julia> a_hat_genus(proj(2))
-1//8
julia> signature(proj(2))
1
julia> signature(complete_intersection(proj(3), 4)) # signature of a K3 surface
-16julia> todd(2)
1 + 1//2*c₁ + 1//12*c₁^2 + 1//12*c₂
julia> a_hat_genus(2)
1 - 1//24*p₁ + 7//5760*p₁^2 - 1//1440*p₂
julia> l_genus(2)
1 + 1//3*p₁ - 1//45*p₁^2 + 7//45*p₂Chow ring
AbstractAlgebra.Generic.basis — Functionbasis(X::AbsVariety)Return an additive basis of the Chow ring of $X$, grouped by increasing degree (i.e., increasing codimension).
basis(k::Int, X::AbsVariety)Return an additive basis of the Chow ring of $X$ in codimension $k$.
Singular.betti — Functionbetti(X::AbsVariety)Return the Betti numbers of the Chow ring of $X$. Note that these are not necessarily equal to the usual Betti numbers, i.e., the dimensions of (co)homologies.
AbstractAlgebra.integral — Methodintegral(x::ChRingElem)Compute the integral of a Chow ring element.
If the variety $X$ has a (unique) point class X.point, the integral will be a number (an fmpq or a function field element). Otherwise the 0-dimensional part of $x$ is returned.
IntersectionTheory.dual_basis — Functiondual_basis(k::Int, X::AbsVariety)Compute the dual basis of the additive basis in codimension $k$ given by basis(k, X) (the returned elements are therefore in codimension $\dim X-k$).
dual_basis(X::AbsVariety)Compute the dual basis with respect to the additive basis given by basis(X), grouped by decreasing degree (i.e., decreasing codimension).
IntersectionTheory.intersection_matrix — Functionintersection_matrix(a::Vector)
intersection_matrix(a::Vector, b::Vector)
intersection_matrix(X::AbsVariety)Compute the intersection matrix among entries of a vector $a$ of Chow ring elements, or between two vectors $a$ and $b$. For a variety $X$, this computes the intersection matrix of the additive basis given by basis(X).
Methods for Grassmannians.
IntersectionTheory.schubert_class — Methodschubert_class(G::AbsVariety, λ::Int...)
schubert_class(G::AbsVariety, λ::AbstractVector)Return the Schubert class $\sigma_\lambda$ on a (relative) Grassmannian $G$.
IntersectionTheory.schubert_classes — Functionschubert_classes(m::Int, G::AbsVariety)Return all the Schubert classes in codimension $m$ on a (relative) Grassmannian $G$.
Examples
julia> basis(proj(2))
3-element Vector{Vector{IntersectionTheory.ChRingElem}}:
[1]
[h]
[h^2]
julia> G = grassmannian(2, 4)
AbsVariety of dim 4
julia> basis(G)
5-element Vector{Vector{IntersectionTheory.ChRingElem}}:
[1]
[c₁]
[c₂, c₁^2]
[c₁*c₂]
[c₂^2]
julia> dual_basis(G)
5-element Vector{Vector{IntersectionTheory.ChRingElem}}:
[c₂^2]
[c₁*c₂]
[-c₁^2 + 2*c₂, c₁^2 - c₂]
[c₁]
[1]
julia> betti(G)
5-element Vector{Int64}:
1
1
2
1
1
julia> basis(2, G)
2-element Vector{IntersectionTheory.ChRingElem}:
c₂
c₁^2
julia> intersection_matrix(basis(2, G))
[1 1]
[1 2]
julia> intersection_matrix(basis(2, G), dual_basis(2, G))
[1 0]
[0 1]
julia> intersection_matrix(G)
[0 0 0 0 0 1]
[0 0 0 0 1 0]
[0 0 1 1 0 0]
[0 0 1 2 0 0]
[0 1 0 0 0 0]
[1 0 0 0 0 0]
julia> schubert_class(G, [1,1])
c₂
julia> schubert_classes(2, G)
2-element Vector{IntersectionTheory.ChRingElem}:
c₁^2 - c₂
c₂Notice that intersection_matrix(X) contains a lot of redundant information: intersection numbers that are not in complementary codimensions are always 0. So usually it is better to separate the components in different codimensions.
Construct new varieties
See also Constructors.
Base.:* — Method*(X::AbsVariety, Y::AbsVariety)Construct the product variety $X\times Y$. If both $X$ and $Y$ have a polarization, $X\times Y$ will be endowed with the polarization of the Segre embedding.
IntersectionTheory.complete_intersection — Functioncomplete_intersection(X::AbsVariety, degs::Int...)
complete_intersection(X::AbsVariety, degs::Vector{Int})Construct the complete intersection in $X$ of general hypersurfaces with degrees $d_1,\dots,d_k$.
IntersectionTheory.section_zero_locus — Functionsection_zero_locus(F::AbsBundle)Construct the zero locus of a general section of a bundle $F$.
Use the argument class=true to only compute the class of the zero locus (same as ctop(F)).
IntersectionTheory.degeneracy_locus — Functiondegeneracy_locus(k::Int, F::AbsBundle, G::AbsBundle)Construct the $k$-degeneracy locus for a general bundle map from $F$ to $G$.
Use the argument class=true to only compute the class of the degeneracy locus.
Examples
julia> proj(1) * proj(1)
AbsVariety of dim 2
julia> complete_intersection(proj(3), 4)
AbsVariety of dim 2
julia> section_zero_locus(OO(proj(3), 4))
AbsVariety of dim 2
julia> X, (A, B) = variety(2, [3=>"a", 4=>"b"]);
julia> D = degeneracy_locus(2, A, B)
AbsVariety of dim 0
julia> pushforward(D → X, D(1))
a₁^2 - a₁*b₁ - a₂ + b₂
julia> degeneracy_locus(2, A, B, class=true)
a₁^2 - a₁*b₁ - a₂ + b₂