Abstract Variety Maps

Types

The OSCAR type for abstract variety maps is AbstractVarietyMap.

Constructors

map — Function

map(X::AbstractVariety, Y::AbstractVariety, fˣ::Vector, fₓ = nothing; inclusion::Bool = false, symbol::String = "x")

Return an abstract variety map $f:X \rightarrow Y$ by specifying the pullbacks of the generators of the Chow ring $N^*(Y)_{\mathbb Q}.$ If needed, also specify the pushforward map $N^*(X)_{\mathbb Q} \rightarrow N^*(Y)_{\mathbb Q}.$

Note

The pullback is relatively easy to describe since it is functorial, while the pushforward is usually more complicated. In some cases, the pushforward of an element $x \in N^*(X)_{\mathbb Q}$ can be automatically computed via pullback. Specifically, the projection formula tells us that

\[f_\ast(f^\ast(y)\cdot x) = f_\ast(x)\cdot y \;\text{ for all }\; x\in N^*(X)_{\mathbb Q}, y\in N^*(Y)_{\mathbb Q}.\]

Since we are working with numerical equivalence, to determine $f_\ast(x)$, it suffices to know all its intersection numbers, which can readily be computed via pullback, provided that all classes in $N^*(Y)_{\mathbb Q}$ are known (or at least those classes having non-zero intersection numbers with $f_\ast(x)$). This is the case in the following situations:

When $Y$ is a point or a curve.
When all classes in $N^*(Y)_{\mathbb Q}$ are known.
When :alg is passed as the fourth argument. This can be done when we are certain that the computed pushforward is correct, even though not all classes in $N^*(X)_{\mathbb Q}$ are known.

In the other cases, if no pushforward map has been specified, a warning will be given when trying to do pushforward.

Note

In the case of an inclusion X $\hookrightarrow$ Y where the class of X in $N^*(Y)_{\mathbb Q}$ is not known, use the argument extend_inclusion = true. Then, a modified version of Y will be created, with extra classes added so that one can pushforward all classes on X. See the subsection Example: Cubic fourfolds of the documentation for an example.

Examples

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

julia> k, l = gens(P2xP2)
2-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 k
 l

julia> P8 = abstract_projective_space(8)
AbstractVariety of dim 8

julia> h = gens(P8)[1]
h

julia> Se = map(P2xP2, P8, [k+l]) # Segre embedding
AbstractVarietyMap from AbstractVariety of dim 4 to AbstractVariety of dim 8

julia> pullback(Se, h)
k + l

julia> pushforward(Se, k+l)
6*h^5

julia> pushforward(Se, pullback(Se, h)*(k+l)) == h*pushforward(Se, k+l)
true

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

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identity_map — Method

identity_map(X::AbstractVariety)

Return the identity map on X.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Underlying Data of an Abstract Variety Map

An abstract variety map is made up from (a selection of) the data discussed here:

domain — Method

 domain(f::AbstractVarietyMap)

Return the domain of f.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

codomain — Method

 codomain(f::AbstractVarietyMap)

Return the codomain of f.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

dim — Method

dim(f::AbstractVarietyMap)

Return the relative dimension of f, that is, return dim(domain(f)) - dim(codomain(f)).

Examples

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

julia> P5 = abstract_projective_space(5, symbol = "H")
AbstractVariety of dim 5

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

julia> i = map(P2, P5, [2*h])
AbstractVarietyMap from AbstractVariety of dim 2 to AbstractVariety of dim 5

julia> dim(i)
-3

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

pullback — Method

pullback(f::AbstractVarietyMap, y::MPolyDecRingOrQuoElem)

Return the pullback of y via f.

Examples

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

julia> P5 = abstract_projective_space(5, symbol = "H")
AbstractVariety of dim 5

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

julia> i = map(P2, P5, [2*h])
AbstractVarietyMap from AbstractVariety of dim 2 to AbstractVariety of dim 5

julia> H = gens(P5)[1]
H

julia> pullback(i, H)
2*h

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

pushforward — Method

pushforward(f::AbstractVarietyMap, x::MPolyDecRingOrQuoElem)

Return the pushforward of x via f.

Examples

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

julia> P5 = abstract_projective_space(5, symbol = "H")
AbstractVariety of dim 5

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

julia> i = map(P2, P5, [2*h])
AbstractVarietyMap from AbstractVariety of dim 2 to AbstractVariety of dim 5

julia> pushforward(i, h)
2*H^4

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

tangent_bundle — Method

tangent_bundle(f::AbstractVarietyMap)

If domain(f) and codomain(f) are given tangent bundles, return the relative tangent bundle 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 = projective_bundle(T)
AbstractVariety of dim 3

julia> pr = structure_map(PT)
AbstractVarietyMap from AbstractVariety of dim 3 to AbstractVariety of dim 2

julia> PBT = pullback(pr, T)
AbstractBundle of rank 2 on AbstractVariety of dim 3

julia> tangent_bundle(PT) - PBT == tangent_bundle(pr)
true

julia> PBT*OO(PT, 1) - OO(PT) == tangent_bundle(pr) # relative Euler sequence
true

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Further Data Associated to an Abstract Variety Map

cotangent_bundle — Method

cotangent_bundle(f::AbstractVarietyMap)

If domain(f) and codomain(f) are given tangent bundles, return the relative cotangent bundle of f.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

todd_class — Method

todd_class(f::AbstractVarietyMap)

Return the Todd class of the relative tangent bundle of f.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Operations on Abstract Variety Maps

compose — Method

compose(f::AbstractVarietyMap, g::AbstractVarietyMap)

Given abstract variety maps f : X $\to$ Y and g : Y $\to$ Z, say, return their composition.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source