Abstract Varieties

 abstract_variety(n::Int, A::Union{MPolyDecRing, MPolyQuoRing{<:MPolyDecRingElem}})

Return an abstract variety of dimension n with Chow ring A.

Examples

julia> R, (h,) = graded_polynomial_ring(QQ, [:h])
(Graded multivariate polynomial ring in 1 variable over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[h])

julia> A, _ = quo(R, ideal(R, [h^3]))
(Quotient of multivariate polynomial ring by ideal (h^3), Map: R -> A)

julia> P2 = abstract_variety(2, A)
AbstractVariety of dim 2

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> TP2 = tangent_bundle(P2)
AbstractBundle of rank 2 on AbstractVariety of dim 2

julia> chern_character(TP2)
3//2*h^2 + 3*h + 2

abstract_point()

Return an abstract variety consisting of a point.

Examples

julia> p = abstract_point()
AbstractVariety of dim 0

julia> chow_ring(p)
Quotient
  of multivariate polynomial ring in 1 variable over QQ graded by
    p -> [1]
  by ideal (p)

abstract_projective_space(n::Int; base::Ring = QQ, symbol::String = "h")

Return the abstract projective space of lines in an n+1-dimensional vector space.

Examples

julia> P3 = abstract_projective_space(3)
AbstractVariety of dim 3

julia> chow_ring(P3)
Quotient
  of multivariate polynomial ring in 1 variable over QQ graded by
    h -> [1]
  by ideal (h^4)

julia> T, (s,t) = polynomial_ring(QQ, [:s, :t])
(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[s, t])

julia> QT = fraction_field(T)
Fraction field
  of multivariate polynomial ring in 2 variables over QQ

julia> P3 = abstract_projective_space(3, base = QT)
AbstractVariety of dim 3

julia> chow_ring(P3)
Quotient
  of multivariate polynomial ring in 1 variable over QT graded by
    h -> [1]
  by ideal (h^4)

julia> TB = tangent_bundle(P3)
AbstractBundle of rank 3 on AbstractVariety of dim 3

julia> CTB = cotangent_bundle(P3)
AbstractBundle of rank 3 on AbstractVariety of dim 3

julia> chern_character((s*TB)*(t*CTB))
-4*s*t*h^2 + 9*s*t

abstract_grassmannian(k::Int, n::Int; base::Ring = QQ, symbol::String = "c")

Return the abstract Grassmannian $\mathrm{G}(k, n)$ of k-dimensional subspaces of an n-dimensional vector space.

Examples

julia> G = abstract_grassmannian(2,4)
AbstractVariety of dim 4

julia> CR = chow_ring(G)
Quotient
  of multivariate polynomial ring in 2 variables over QQ graded by
    c[1] -> [1]
    c[2] -> [2]
  by ideal (-c[1]^3 + 2*c[1]*c[2], c[1]^4 - 3*c[1]^2*c[2] + c[2]^2)

julia> S = tautological_bundles(G)[1]
AbstractBundle of rank 2 on AbstractVariety of dim 4

julia> V = [chern_class(S, i) for i = 1:2]
2-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 c[1]
 c[2]

julia> is_regular_sequence(gens(modulus(CR)))
true

julia> Q = tautological_bundles(G)[2]
AbstractBundle of rank 2 on AbstractVariety of dim 4

julia> tangent_bundle(G) == dual(S)*Q
true

abstract_flag_variety(dims::Int...; base::Ring = QQ, symbol::String = "c")
abstract_flag_variety(dims::Vector{Int}; base::Ring = QQ, symbol::String = "c")

Given integers, say, $d_1, \dots, d_{k}, n$ with $0 < d_1 < \dots < d_{k} < n$ or a vector of such integers, return the abstract flag variety $\mathrm{F}(d_1, \dots, d_{k}; n)$ of nested sequences of subspaces of dimensions $d_1, \dots, d_{k}$ of an $n$-dimensional vector space.

Examples

julia> F = abstract_flag_variety(1,3,4)
AbstractVariety of dim 5

julia> chow_ring(F)
Quotient
  of multivariate polynomial ring in 4 variables over QQ graded by
    c[1, 1] -> [1]
    c[2, 1] -> [1]
    c[2, 2] -> [2]
    c[3, 1] -> [1]
  by ideal with 4 generators

julia> modulus(chow_ring(F))
Ideal generated by
  -c[1, 1]*c[2, 2]*c[3, 1]
  -c[1, 1]*c[2, 1]*c[3, 1] - c[1, 1]*c[2, 2] - c[2, 2]*c[3, 1]
  -c[1, 1]*c[2, 1] - c[1, 1]*c[3, 1] - c[2, 1]*c[3, 1] - c[2, 2]
  -c[1, 1] - c[2, 1] - c[3, 1]

complete_intersection(X::AbstractVariety, degs::Int...)
complete_intersection(X::AbstractVariety, degs::Vector{Int})

Return the complete intersection in X of general hypersurfaces with the given degrees.

Examples

julia> P3 = abstract_projective_space(3)
AbstractVariety of dim 3

julia> CI = complete_intersection(P3, 2, 2)
AbstractVariety of dim 1

julia> dim(CI)
1

julia> degree(CI)
4

julia> chow_ring(CI)
Quotient
  of multivariate polynomial ring in 1 variable over QQ graded by
    h -> [1]
  by ideal (h^2)

abstract_projective_bundle(F::AbstractBundle; symbol::String = "z")

Return the projective bundle of 1-dimensional subspaces in the fibers of F.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> T = tangent_bundle(P2)
AbstractBundle of rank 2 on AbstractVariety of dim 2

julia> PT = abstract_projective_bundle(T)
AbstractVariety of dim 3

julia> chow_ring(PT)
Quotient
  of multivariate polynomial ring in 2 variables over QQ graded by
    z -> [1]
    h -> [1]
  by ideal (h^3, z^2 + 3*z*h + 3*h^2)

julia> [chern_class(T, i) for i = 1:2]
2-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 3*h
 3*h^2

julia> gens(PT)[1]
z

julia> gens(PT)[1] == hyperplane_class(PT)
true

julia> G = abstract_grassmannian(3, 5)
AbstractVariety of dim 6

julia> USBd = dual(tautological_bundles(G)[1])
AbstractBundle of rank 3 on AbstractVariety of dim 6

julia> F = symmetric_power(USBd, 2)
AbstractBundle of rank 6 on AbstractVariety of dim 6

julia> PF = abstract_projective_bundle(F)
AbstractVariety of dim 11

julia> A = symmetric_power(USBd, 5) - symmetric_power(USBd, 3)*OO(PF, -1)
AbstractBundle of rank 11 on AbstractVariety of dim 11

julia> integral(top_chern_class(A))
609250

abstract_hirzebruch_surface(n::Int)

Return the n-th Hirzebruch surface.

Examples

julia> H2 =  abstract_hirzebruch_surface(2)
AbstractVariety of dim 2

julia> chow_ring(H2)
Quotient
  of multivariate polynomial ring in 2 variables over QQ graded by
    z -> [1]
    h -> [1]
  by ideal (h^2, z^2 - 2*z*h)

abstract_flag_bundle(F::AbstractBundle, dims::Int...; symbol::String = "c")
abstract_flag_bundle(F::AbstractBundle, dims::Vector{Int}; symbol::String = "c")

Given integers, say, $d_1, \dots, d_{k}, n$ with $0 < d_1 < \dots < d_{k} < n$ or a vector of such integers, and given an abstract bundle $F$ of rank $n$, return the abstract flag bundle of nested sequences of subspaces of dimensions $d_1, \dots, d_{k}$ in the fibers of $F$.

Examples

julia> P = abstract_projective_space(4)
AbstractVariety of dim 4

julia> F = exterior_power(cotangent_bundle(P),  3)*OO(P,3)
AbstractBundle of rank 4 on AbstractVariety of dim 4

julia> FB = abstract_flag_bundle(F, 1, 3)
AbstractVariety of dim 9

julia> CR = chow_ring(FB)
Quotient
  of multivariate polynomial ring in 5 variables over QQ graded by
    c[1, 1] -> [1]
    c[2, 1] -> [1]
    c[2, 2] -> [2]
    c[3, 1] -> [1]
    h -> [1]
  by ideal with 5 generators

julia> modulus(CR)
Ideal generated by
  h^5
  -c[1, 1]*c[2, 2]*c[3, 1] + h^4
  -c[1, 1]*c[2, 1]*c[3, 1] - c[1, 1]*c[2, 2] - c[2, 2]*c[3, 1] - 2*h^3
  -c[1, 1]*c[2, 1] - c[1, 1]*c[3, 1] - c[2, 1]*c[3, 1] - c[2, 2] + 4*h^2
  -c[1, 1] - c[2, 1] - c[3, 1] - 3*h

julia> tautological_bundles(FB)
3-element Vector{AbstractBundle}:
 AbstractBundle of rank 1 on AbstractVariety of dim 9
 AbstractBundle of rank 2 on AbstractVariety of dim 9
 AbstractBundle of rank 1 on AbstractVariety of dim 9

zero_locus_section(F::AbstractBundle; class::Bool = false)

Return the zero locus of a general section of F.

Use the argument class = true to only compute the class of the zero locus (same as top_chern_class(F)).

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> C = zero_locus_section(OO(P2, 3)) # a plane cubic curve
AbstractVariety of dim 1

julia> dim(C)
1

julia> degree(C)
3

julia> P3 = abstract_projective_space(3)
AbstractVariety of dim 3

julia> C = zero_locus_section(OO(P3, 2) + OO(P3, 2)) # a complete intersection
AbstractVariety of dim 1

julia> dim(C)
1

julia> degree(C)
4

julia> P4 = abstract_projective_space(4)
AbstractVariety of dim 4

julia> h = gens(P4)[1]
h

julia> F = abstract_bundle(P4, 2, 10*h^2 + 5*h + 1) # Horrocks-Mumford bundle
AbstractBundle of rank 2 on AbstractVariety of dim 4

julia> A = zero_locus_section(F) # abelian surface
AbstractVariety of dim 2

julia> dim(A)
2

julia> degree(A)
10

degeneracy_locus(F::AbstractBundle, G::AbstractBundle, k::Int; class::Bool=false)

Return the k-th degeneracy locus of a general map from F to G.

Use the argument class = true to only compute the class of the degeneracy locus (Porteous' formula).

Examples

julia> P4 = abstract_projective_space(4)
AbstractVariety of dim 4

julia> F = 3*OO(P4, -1)
AbstractBundle of rank 3 on AbstractVariety of dim 4

julia> G = cotangent_bundle(P4)*OO(P4,1)
AbstractBundle of rank 4 on AbstractVariety of dim 4

julia> CZ = degeneracy_locus(F, G, 2, class = true) # only class of degeneracy locus
4*h^2

julia> CZ == chern_class(G-F, 2) # Porteous' formula
true

julia> Z = degeneracy_locus(F, G, 2) # Veronese surface in P4
AbstractVariety of dim 2

julia> degree(Z)
4

julia> P = abstract_projective_space(4, symbol = "H")
AbstractVariety of dim 4

julia> F = exterior_power(cotangent_bundle(P), 3)*OO(P,3)
AbstractBundle of rank 4 on AbstractVariety of dim 4

julia> G = OO(P, 1)+4*OO(P)
AbstractBundle of rank 5 on AbstractVariety of dim 4

julia> Z = degeneracy_locus(F, G, 3) # rational surface in P4
AbstractVariety of dim 2

julia> K = canonical_class(Z)
z - H

julia> integral(K^2)
-7

julia> H = hyperplane_class(Z)
H

julia> integral(H^2) # degree of surface
8

julia> A = K+H
z

julia> integral(A^2) # degree of first adjoint surface which is a Del Pezzo surface in P5
5

 dim(X::AbstractVariety)

Return the dimension of X.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> P3 = abstract_projective_space(3)
AbstractVariety of dim 3

julia> dim(P2*P3)
5

chow_ring(X::AbstractVariety)

Return the Chow ring of X.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> P3 = abstract_projective_space(3, symbol = "H")
AbstractVariety of dim 3

julia> chow_ring(P2*P3)
Quotient
  of multivariate polynomial ring in 2 variables over QQ graded by
    h -> [1]
    H -> [1]
  by ideal (h^3, H^4)

base(X::AbstractVariety)

Return the coefficient ring of the polynomial ring underlying the Chow ring of X.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> chow_ring(P2)
Quotient
  of multivariate polynomial ring in 1 variable over QQ graded by
    h -> [1]
  by ideal (h^3)

julia> base(P2)
Rational field

point_class(X::AbstractVariety)

If X has been given a point class, return that class.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> P3 = abstract_projective_space(3, symbol = "H")
AbstractVariety of dim 3

julia> p = point_class(P2*P3)
h^2*H^3

julia> degree(p)
[5]

julia> integral(p)
1

tangent_bundle(X::AbstractVariety)

If X has been given a tangent bundle, return that bundle.

Examples

julia> G = abstract_grassmannian(2,5)
AbstractVariety of dim 6

julia> TG = tangent_bundle(G)
AbstractBundle of rank 6 on AbstractVariety of dim 6

julia> chern_character(TG)
-5//6*c[1]^3 + 5//24*c[1]^2*c[2] + 3//2*c[1]^2 + 5//2*c[1]*c[2] - 5*c[1] + 7//72*c[2]^3 - 25//24*c[2]^2 - c[2] + 6

hyperplane_class(X::AbstractVariety)

If X has been given the class of a hyperplane section of X, return that class.

Examples

julia> G = abstract_grassmannian(2, 5)
AbstractVariety of dim 6

julia> hyperplane_class(G)
-c[1]

julia> degree(G) == integral(hyperplane_class(G)^dim(G)) == 5
true

julia> P1s = abstract_projective_space(1, symbol = "s")
AbstractVariety of dim 1

julia> P1t = abstract_projective_space(1, symbol = "t")
AbstractVariety of dim 1

julia> P = P1s*P1t
AbstractVariety of dim 2

julia> H = hyperplane_class(P)
s + t

julia> integral(H^dim(P))
2

tautological_bundles(X::AbstractVariety)

If X has been given tautological bundles, return these bundles.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> TB = tautological_bundles(P2)
2-element Vector{AbstractBundle}:
 AbstractBundle of rank 1 on AbstractVariety of dim 2
 AbstractBundle of rank 2 on AbstractVariety of dim 2

julia> TB[1] == OO(P2, -1)
true

julia> TB[2] == tangent_bundle(P2)*OO(P2, -1)
true

julia> G = abstract_grassmannian(3, 5)
AbstractVariety of dim 6

julia> tautological_bundles(G)
2-element Vector{AbstractBundle}:
 AbstractBundle of rank 3 on AbstractVariety of dim 6
 AbstractBundle of rank 2 on AbstractVariety of dim 6

structure_map(X::AbstractVariety)

If X has been given a structure map, return that map.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> structure_map(P2)
AbstractVarietyMap from AbstractVariety of dim 2 to AbstractVariety of dim 0

julia> T = tangent_bundle(P2)
AbstractBundle of rank 2 on AbstractVariety of dim 2

julia> E = abstract_projective_bundle(T, symbol = "H")
AbstractVariety of dim 3

julia> chow_ring(E)
Quotient
  of multivariate polynomial ring in 2 variables over QQ graded by
    H -> [1]
    h -> [1]
  by ideal (h^3, H^2 + 3*H*h + 3*h^2)

julia> structure_map(E)
AbstractVarietyMap from AbstractVariety of dim 3 to AbstractVariety of dim 2

trivial_line_bundle(X::AbstractVariety)

Return the trivial line bundle $\mathcal O_X$ on X. Alternatively, use OO instead of trivial_line_bundle.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> OX = trivial_line_bundle(P2)
AbstractBundle of rank 1 on AbstractVariety of dim 2

julia> chern_character(OX)
1

line_bundle(X::AbstractVariety, n::RingElement)

If X has been given a hyperplane class, return the line bundle $\mathcal O_X(n)$ on X.

line_bundle(X::AbstractVariety, D::Union{MPolyDecRingElem, MPolyQuoRingElem})

Given an element D of the Chow ring of X, return the line bundle $\mathcal O_X(D)$ with first Chern class $D[1]$. Here, $D[1]$ is the degree-1 part of D.

Alternatively, use OO instead of line_bundle.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> tautological_bundles(P2)[1] == OO(P2, -1)
true

julia> h = gens(P2)[1]
h

julia> OO(P2, h) == OO(P2, 1)
true

cotangent_bundle(X::AbstractVariety)

If X has been given a tangent bundle, return the dual bundle.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> cotangent_bundle(P2) == dual(tangent_bundle(P2)) 
true

canonical_class(X::AbstractVariety)

If X has been given a tangent bundle, return the canonical class of X.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> canonical_class(P2) == chern_class(cotangent_bundle(P2), 1)
true

julia> canonical_class(P2) 
-3*h

canonical_bundle(X::AbstractVariety)

If X has been given a tangent bundle, return the canonical bundle on X.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> canonical_bundle(P2) == det(cotangent_bundle(P2))
true

degree(X::AbstractVariety)

If X has been given a hyperplane class, return the corresponding degree of X.

Examples

julia> G = abstract_grassmannian(2,5)
AbstractVariety of dim 6

julia> degree(G)
5

hilbert_polynomial(X::AbstractVariety)

If X has been given a hyperplane class, return the corresponding Hilbert polynomial of X.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> hilbert_polynomial(P2) == hilbert_polynomial(OO(P2))
true

julia> hilbert_polynomial(P2)
1//2*t^2 + 3//2*t + 1

basis(X::AbstractVariety)
basis(X::AbstractVariety, k::Int)

If K = base(X), return a K-basis of the Chow ring of X (return the elements of degree k in that basis).

Examples

julia> G = abstract_grassmannian(2,4)
AbstractVariety of dim 4

julia> chow_ring(G)
Quotient
  of multivariate polynomial ring in 2 variables over QQ graded by
    c[1] -> [1]
    c[2] -> [2]
  by ideal (-c[1]^3 + 2*c[1]*c[2], c[1]^4 - 3*c[1]^2*c[2] + c[2]^2)

julia> basis(G)
5-element Vector{Vector{MPolyQuoRingElem}}:
 [1]
 [c[1]]
 [c[2], c[1]^2]
 [c[1]*c[2]]
 [c[2]^2]

julia> basis(G, 2)
2-element Vector{MPolyQuoRingElem}:
 c[2]
 c[1]^2

betti_numbers(X::AbstractVariety)

Return the Betti numbers of the Chow ring of X.

Examples

julia> P2xP2 = abstract_projective_space(2, symbol = "k")*abstract_projective_space(2, symbol = "l")
AbstractVariety of dim 4

julia> betti_numbers(P2xP2)
5-element Vector{Int64}:
 1
 2
 3
 2
 1

julia> basis(P2xP2)
5-element Vector{Vector{MPolyQuoRingElem}}:
 [1]
 [l, k]
 [l^2, k*l, k^2]
 [k*l^2, k^2*l]
 [k^2*l^2]

intersection_matrix(X::AbstractVariety)

If b = vcat(basis(X)...), return matrix([integral(bi*bj) for bi in b, bj in b]).

intersection_matrix(a::Vector, b::Vector)

Return matrix([integral(ai*bj) for ai in a, bj in b]).

intersection_matrix(a::Vector)

As above, with b = a.

Examples

julia> G = abstract_grassmannian(2,4)
AbstractVariety of dim 4

julia> basis(G)
5-element Vector{Vector{MPolyQuoRingElem}}:
 [1]
 [c[1]]
 [c[2], c[1]^2]
 [c[1]*c[2]]
 [c[2]^2]

julia> b = vcat(basis(G)...)
6-element Vector{MPolyQuoRingElem}:
 1
 c[1]
 c[2]
 c[1]^2
 c[1]*c[2]
 c[2]^2

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> integral(b[4]*b[4])
2

dual_basis(X::AbstractVariety)
dual_basis(X::AbstractVariety, k::Int)

If K = base(X), return a K-basis of the Chow ring of X which is dual to basis(X) with respect to the K-bilinear form defined by intersection_matrix(X) (return the elements of degree k in the dual basis).

Examples

julia> G = abstract_grassmannian(2,4)
AbstractVariety of dim 4

julia> b = basis(G)
5-element Vector{Vector{MPolyQuoRingElem}}:
 [1]
 [c[1]]
 [c[2], c[1]^2]
 [c[1]*c[2]]
 [c[2]^2]

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> bd = dual_basis(G)
5-element Vector{Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}}:
 [c[2]^2]
 [c[1]*c[2]]
 [-c[1]^2 + 2*c[2], c[1]^2 - c[2]]
 [c[1]]
 [1]

julia> integral(b[3][2]*b[3][2])
2

julia> integral(b[3][2]*bd[3][2])
1

euler_number(X::AbstractVariety)

Return the Euler number of X.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> euler_number(P2)
3

julia> euler_number(P2) == integral(total_chern_class(tangent_bundle(P2)))
true

product(X::AbstractVariety, Y::AbstractVariety)

Return the product $X\times Y$. Alternatively, use *.

julia> P2 = abstract_projective_space(2);

julia> P3 = abstract_projective_space(3, symbol = "H");

julia> P2xP3 = P2*P3
AbstractVariety of dim 5

julia> chow_ring(P2xP3)
Quotient
  of multivariate polynomial ring in 2 variables over QQ graded by
    h -> [1]
    H -> [1]
  by ideal (h^3, H^4)