inclusion -- build the freest possible inclusion map

Synopsis

Usage:

inclusion(f, NormalClass => c, Codimension => r, SuperTangent => tY, SuperDimension => dY, Base => S)

inclusion(f, SubTangent => tX, SubDimension => dX, SuperTangent => tY, SuperDimension => dY)

inclusion(f)
Inputs:
- f, from A to B; we think of A as part of the Chow ring of Y and B as part of the Chow ring of X
Optional inputs:
- Codimension => an integer, default value null, the codimension of X in Y
- SubDimension => an integer, default value null, the dimension of X
- SuperDimension => an integer, default value null, the dimension of Y
- SubTangent => a ring element, default value null, element of B, the Chern class of the tangent bundle of X
- SuperTangent => a ring element, default value null, element of A, the Chern class of the tangent bundle of Y
- NormalClass => a ring element, default value null, element of B, the Chern class of the normal bundle of X in Y
- Base => a thing, default value null, a Ring or AbstractVariety, the base ring/variety to work over
Outputs:
- an abstract variety map, the natural map from X to Z.
Consequences:
- An extensionAlgebra of A by B is built, with normal class equal to the degree-r part of c. This algebra is made into the ring of a variety Z, see abstractVariety. A tangent class is installed on Z, see tangentBundle. If A and B each have an integral defined, then an integral is defined on the coordinate ring of Z, and a StructureMap is installed on Z.
- A variety X for B is built if B is not already the ring of a variety, see abstractVariety. A tangent class is installed on X if it does not already have one, see tangentBundle. A StructureMap is installed if B already has an integral defined.
- If Base option is used, the supplied ring is made into the ring of a variety, if it is not already one.

Description

Given the pullback map f from A to B, builds the freest possible extension E of A by B (see extensionAlgebra), and then adds appropriate metadata to make the maps from E to B and vice-versa into an AbstractVarietyMap. Enough information must be given to compute the dimensions of X and Y, either by using the SubDimension, SuperDimension, and Codimension options, or by having varieties already attached to A and/or B. Likewise, enough information must be given to compute the tangent classes of X and Y.

This construction is useful for computations where the pullback map is known but the pushforward is either not known or cannot be defined.

i1 : p = point

o1 = point

o1 : an abstract variety of dimension 0

i2 : S = intersectionRing p

o2 = S

o2 : PolynomialRing

i3 : Y = projectiveBundle(5,p)

o3 = Y

o3 : a flag bundle with subquotient ranks {1, 5}

i4 : A = intersectionRing Y

o4 = A

o4 : QuotientRing

i5 : B = S[h, Join => false]/h^3 -- A^*(P2), but using 2 times a line as the generating class:

o5 = B

o5 : QuotientRing

i6 : integral B := (b) -> (4 * coefficient((B_0)^2, b))

o6 = FunctionClosure[stdio:6:14-6:50]

o6 : FunctionClosure

i7 : c = 1 + (9/2)*h + (15/2)*h^2 -- normal class

     15 2   9
o7 = --h  + -h + 1
      2     2

o7 : B

i8 : f = map(B,A,{-h, h, h^2, h^3, h^4, h^5})

                         2
o8 = map (B, A, {-h, h, h , 0, 0, 0})

o8 : RingMap B <-- A

i9 : i = inclusion(f,
         NormalClass => c,
         Codimension => 3,
         Base => p) -- Base not necessary, will be correctly computed

o9 = i

o9 : a map to a variety from a variety

i10 : Z = target i

o10 = Z

o10 : an abstract variety of dimension 5

i11 : X = source i

o11 = X

o11 : an abstract variety of dimension 2

i12 : Xstruct = X / point

o12 = Xstruct

o12 : a map to point from X

i13 : rank Xstruct_* tangentBundle X

o13 = 8

i14 : integral chern tangentBundle Z

o14 = 6

o14 : S

Ways to use inclusion:

inclusion(RingMap)

For the programmer

The object inclusion is a method function with options.