Cycles and divisors

AbsAlgebraicCycle{CoveredSchemeType<:AbsCoveredScheme, CoefficientRingType<:Ring}

An algebraic cycle $D$ on a (locally) Noetherian integral scheme $X$ with coefficients in a ring $R$ is a formal linear combination $\sum_i a_i D_i$ with $D_i \subseteq X$ integral, closed subschemes and the $a_i \in R$.

Such a cycle is represented non-uniquely as a formal sum $E = \sum_l b_l \mathcal{I}_l$ of equidimensional ideal sheaves $\mathcal{I}_l \subseteq \mathcal{O}_X$. For an equidimensional ideal sheaf $\mathcal{I}$ its interpretation as a cycle is as follows: Let $V(\mathcal{I})=E_{1} \cup \dots \cup E_{n}$ be the decomposition of the vanishing locus of $\mathcal{I}$ into irreducible components $E_i=V(\mathcal{P}_i)$ with $\mathcal{P}_i$ prime. Then $E$ corresponds to the cycle $D = \sum_{i=1}^{n} \mathrm{colength}_{\mathcal{P}_i}(\mathcal{I})E_i$.

Examples

julia> P2 = projective_space(QQ,2); (s0,s1,s2) = homogeneous_coordinates(P2);

julia> I = ideal_sheaf(P2,ideal([s0,s1^2]))
Sheaf of ideals
  on scheme over QQ covered with 3 patches
    1: [(s1//s0), (s2//s0)]   affine 2-space
    2: [(s0//s1), (s2//s1)]   affine 2-space
    3: [(s0//s2), (s1//s2)]   affine 2-space
with restrictions
  1: Ideal (1, (s1//s0)^2)
  2: Ideal ((s0//s1), 1)
  3: Ideal ((s0//s2), (s1//s2)^2)

julia> D = algebraic_cycle(I)
Effective algebraic cycle
  on scheme over QQ covered with 3 patches
with coefficients in integer ring
given as the formal sum of
  1 * sheaf of ideals

julia> irreducible_decomposition(D)
Effective algebraic cycle
  on scheme over QQ covered with 3 patches
with coefficients in integer ring
given as the formal sum of
  2 * sheaf of prime ideals

algebraic_cycle(X::AbsCoveredScheme, R::Ring) -> AlgebraicCycle

Return the zero AlgebraicCycle over X with coefficients in R.

Examples

julia> P, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> I = ideal([x^3-y^2*z]);

julia> Y = proj(P, I);

julia> Ycov = covered_scheme(Y);

julia> R = ZZ;

julia> algebraic_cycle(Ycov, R)
Zero algebraic cycle
  on scheme over QQ covered with 3 patches
with coefficients in integer ring

algebraic_cycle(I::AbsIdealSheaf, R::Ring) -> AlgebraicCycle

Return the AlgebraicCycle $D = 1 ⋅ I$ with coefficients in $R$ for a sheaf of equidimensional ideals $I$.

Note that $I$ must be equidimensional.

Examples

julia> P, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> I = ideal([x^3-y^2*z]);

julia> Y = proj(P);

julia> II = IdealSheaf(Y, I);

julia> R = ZZ;

julia> algebraic_cycle(II, R)
Effective algebraic cycle
  on scheme over QQ covered with 3 patches
with coefficients in integer ring
given as the formal sum of
  1 * sheaf of ideals

algebraic_cycle(I::AbsIdealSheaf) -> AlgebraicCycle

Return the AlgebraicCycle $D = 1 ⋅ I$ with coefficients in $ℤ$ for a sheaf of equidimensional ideals $I$.

Examples

julia> P, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> I = ideal([x^3-y^2*z]);

julia> Y = proj(P);

julia> II = IdealSheaf(Y, I);

julia> R = ZZ;

julia> algebraic_cycle(II, R)
Effective algebraic cycle
  on scheme over QQ covered with 3 patches
with coefficients in integer ring
given as the formal sum of
  1 * sheaf of ideals

ambient_scheme(D::AbsAlgebraicCycle)

Return the CoveredScheme $X$ on which D is defined.

components(D::AbsAlgebraicCycle)

Return a list of ideal sheaves such that D is a linear combination of the corresponding cycles.

dim(D::AbsAlgebraicCycle)

Return the dimension of the support of the cycle D.

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.

integral(W::AbsAlgebraicCycle)

Assume $W$ is an algebraic cycle on $X$. This returns the sum of the lengths of all the components of dimension 0 of $W$.

is_effective(A::AbsAlgebraicCycle)

Return whether all the coefficients are non-negative.

is_prime(D::AbsAlgebraicCycle)

An algebraic cycle is called prime if it consists of a single irreducible subvariety.

Note that this property is not stable under base extension.

Base.:<=(A::AbsAlgebraicCycle, B::AbsAlgebraicCycle)

\[A \leq B\]

if and only if $B - A$ is effective.

AbsWeilDivisor{CoveredSchemeType, CoefficientRingType} <: AbsAlgebraicCycle{CoveredSchemeType, CoefficientRingType}

A Weil divisor with coefficients of type CoefficientRingType on a (locally) Noetherian integral scheme $X$ of type CoveredSchemeType.

Examples

julia> P2 = projective_space(QQ,2); (s0,s1,s2) = homogeneous_coordinates(P2);

julia> I = ideal((s0*s1)^2);

julia> II = ideal_sheaf(P2, I);

julia> D = weil_divisor(II)
Effective weil divisor
  on scheme over QQ covered with 3 patches
with coefficients in integer ring
given as the formal sum of
  1 * sheaf of ideals

julia> E = irreducible_decomposition(D)
Effective weil divisor
  on scheme over QQ covered with 3 patches
with coefficients in integer ring
given as the formal sum of
  2 * prime ideal sheaf on scheme over QQ covered with 3 patches extended from ideal ((s1//s0)) on affine 2-space
  2 * prime ideal sheaf on scheme over QQ covered with 3 patches extended from ideal ((s0//s1)) on affine 2-space

julia> P = components(E)[1]
Prime ideal sheaf on Scheme over QQ covered with 3 patches extended from Ideal ((s1//s0)) on Affine 2-space

julia> components(D)[1] == II
true

julia> D[II] # to get the coefficient
1

julia> E[P]
2

weil_divisor(X::AbsCoveredScheme, R::Ring) -> WeilDivisor

Return the zero weil divisor on X with coefficients in the ring R.

Examples

julia> P, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> I = ideal([x^3-y^2*z]);

julia> Y = proj(P, I);

julia> Ycov = covered_scheme(Y);

julia> weil_divisor(Ycov, QQ)
Zero weil divisor
  on scheme over QQ covered with 3 patches
with coefficients in rational field

weil_divisor(I::AbsIdealSheaf) -> WeilDivisor

Given an ideal sheaf I of pure codimension $1$, return the weil divisor $D = 1 ⋅ I$ with coefficients in the integer ring.

Example

julia> P, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> Y = proj(P);

julia> I = ideal([(x^3-y^2*z)]);

julia> II = IdealSheaf(Y, I);

julia> weil_divisor(II)
Effective weil divisor
  on scheme over QQ covered with 3 patches
with coefficients in integer ring
given as the formal sum of
  1 * sheaf of ideals
  
julia> JJ = II^2;

julia> D = weil_divisor(JJ)
Effective weil divisor
  on scheme over QQ covered with 3 patches
with coefficients in integer ring
given as the formal sum of
  1 * product of 2 ideal sheaves

julia> irreducible_decomposition(D)
Effective weil divisor
  on scheme over QQ covered with 3 patches
with coefficients in integer ring
given as the formal sum of
  2 * sheaf of prime ideals

weil_divisor(I::AbsIdealSheaf, R::Ring; check::Bool=true)

Given an ideal sheaf I of pure codimension $1$ and a ring R, return the weil divisor $D = 1 ⋅ I$ with coefficients in R.

is_in_linear_system(f::VarietyFunctionFieldElem, D::WeilDivisor; regular_on_complement::Bool=true, check::Bool=true) -> Bool

Return whether the rational function f is in the linear system $|D|$, i.e. if $(f) + D \geq 0$.

Input

• regular_on_complement – set to true if f is regular on the complement of the support of D.

order_of_vanishing(f::VarietyFunctionFieldElem, D::AbsWeilDivisor; check::Bool=true)

Return the order of vanishing of the rational function f on the prime divisor D.

intersect(D::AbsWeilDivisor, E::AbsWeilDivisor; covering::Covering=default_covering(ambient_scheme(D)))

Return the intersection number of the the Weil divisors D and E on a complete smooth surface as defined in [Har77].

Input

The optional keyword argument covering specifies the covering to be used for the computation.

LinearSystem

A linear system of a Weil divisor $D$ on a variety $X$, generated by rational functions $f₁,…,fᵣ ∈ K(X)$.

weil_divisor(L::LinearSystem)

Return the divisor $D$ of the linear system $L = |D|$.

variety(L::LinearSystem)

Return the variety on which L is defined.

subsystem(L::LinearSystem, D::AbsWeilDivisor) -> LinearSystem, MatElem

Given a linear system $L = |E|$ and a divisor $D \leq E$ compute $|D|$ and the matrix representing the inclusion $|D| \hookrightarrow |E|$ with respect to the given bases of both systems.

CartierDivisor{CoveredSchemeType<:AbsCoveredScheme, CoeffType<:RingElem}

A Cartier divisor $C$ on a scheme $X$ with coefficients $a_i$ in a ring $R$ is a formal linear combination $\sum_i a_i D_i$ of effective Cartier divisors $D_i$.

The scheme $X$ is of type CoveredSchemeType. The coefficients $a_i$ are of type CoeffType.

EffectiveCartierDivisor{CoveredSchemeType<:AbsCoveredScheme}

An effective Cartier divisor on a scheme $X$ is a closed subscheme $D \subseteq X$ whose ideal sheaf $\mathcal{I}_D \subseteq \mathcal{O}_X$ is an invertible $\mathcal{O}_X$-module. In particular, $\mathcal{I}_D$ is locally principal.

Internally, $C$ stores a trivializing_covering(C::EffectiveCartierDivisor). The scheme $X$ is of type CoveredSchemeType.

effective_cartier_divisor(I::IdealSheaf;
                          trivializing_covering::Covering = default_covering(scheme(I)))
                                                            -> EffectiveCartierDivisor

Return the effective Cartier divisor defined by the ideal sheaf I, given that I is principal in the given covering of the scheme on which it is defined.

Examples

julia> P, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> I = ideal([x^3-y^2*z]);

julia> Y = proj(P);

julia> II = IdealSheaf(Y, I);

julia> effective_cartier_divisor(II)
Effective cartier divisor
  on scheme over QQ covered with 3 patches
    1: [(y//x), (z//x)]   affine 2-space
    2: [(x//y), (z//y)]   affine 2-space
    3: [(x//z), (y//z)]   affine 2-space
defined by
  sheaf of ideals with restrictions
    1: Ideal (-(y//x)^2*(z//x) + 1)
    2: Ideal ((x//y)^3 - (z//y))
    3: Ideal ((x//z)^3 - (y//z)^2)

effective_cartier_divisor(IP::AbsProjectiveScheme, f::Union{MPolyDecRingElem, MPolyQuoRingElem})

Return the effective Cartier divisor on the projective scheme $X$ defined by the homogeneous polynomial $f$.

cartier_divisor(E::EffectiveCartierDivisor) -> CartierDivisor

Convert an EffectiveCartierDivisor into a CartierDivisor with coefficient $1$ in the ring of integers.

Examples

julia> P, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> I = ideal([x^3-y^2*z]);

julia> Y = proj(P);

julia> II = IdealSheaf(Y, I);

julia> E = effective_cartier_divisor(II)
Effective cartier divisor
  on scheme over QQ covered with 3 patches
    1: [(y//x), (z//x)]   affine 2-space
    2: [(x//y), (z//y)]   affine 2-space
    3: [(x//z), (y//z)]   affine 2-space
defined by
  sheaf of ideals with restrictions
    1: Ideal (-(y//x)^2*(z//x) + 1)
    2: Ideal ((x//y)^3 - (z//y))
    3: Ideal ((x//z)^3 - (y//z)^2)

julia> cartier_divisor(E)
Cartier divisor
  on scheme over QQ covered with 3 patches
with coefficients in integer ring
defined by the formal sum of
  1 * effective cartier divisor on scheme over QQ covered with 3 patches

cartier_divisor(IP::AbsProjectiveScheme, f::Union{MPolyDecRingElem, MPolyQuoRingElem})

Return the (effective) Cartier divisor on the projective scheme $X$ defined by the homogeneous polynomial $f$.

ideal_sheaf(C::EffectiveCartierDivisor)

Return the sheaf of ideals $\mathcal{I}_C \subseteq \mathcal{O}_X$ representing C.

ambient_scheme(C::EffectiveCartierDivisor)

Return the ambient scheme containing C.

ambient_scheme(C::CartierDivisor)

Return the ambient scheme containing C.

coefficient_ring(C::CartierDivisor)

Return the ring of coefficients of C.

components(C::CartierDivisor)

Return a list of effective Cartier divisors $C_i$ such that $C$ is a linear combination of the $C_i$.

trivializing_covering(C::EffectiveCartierDivisor)

Return the trivializing covering of the effective Cartier divisor C.

A covering $(U_i)_{i \in I}$ is called trivializing for $C$ if $C(U_i)$ is principal for all $i \in I$.