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AbstractSheaf -- the class of sheaves given by their Chern classes

Description

This is the class of a sheaf as defined in the package Schubert2. An abstract sheaf is really the data of its base variety (see AbstractVariety), Rank (missing documentation), ChernClass (missing documentation)...

For example, the Horrocks-Mumford bundle on projective 4-space can be represented by the following text. We first compute a “base variety” that is a point pt, and a variable integer named n, in terms of which we can compute the Hilbert polynomial.

i1 : pt = base(n)

o1 = pt

o1 : an abstract variety of dimension 0
i2 : X = projectiveSpace(4,pt,VariableName => h)

o2 = X

o2 : a flag bundle with ranks {4, 1}
i3 : F = abstractSheaf(X, Rank => 2, ChernClass => 1 + 5*h + 10*h^2)

o3 = F

o3 : an abstract sheaf of rank 2 on X
i4 : chi(F**OO_X(n*h))

      1 4   5 3   125 2   125
o4 = --n  + -n  + ---n  + ---n + 2
     12     3      12      6

o4 : QQ[n]
Here in the description of X the number 4 is the rank of the trivial bundle, pt is the base variety of the projective space (in general, we allow projective spaces over an arbitrary base AbstractVariety), and the variable name specifies the variable to use to represent the first Chern Class of the tautological quotient line bundle on the projective space.

See also

Functions and methods returning an abstract sheaf :

  • AbstractSheaf * AbstractSheaf (missing documentation)
  • AbstractSheaf ** AbstractSheaf (missing documentation)

Methods that use an abstract sheaf :

  • - AbstractSheaf (missing documentation)
  • AbstractSheaf + AbstractSheaf (missing documentation)
  • AbstractSheaf + ZZ (missing documentation)
  • AbstractSheaf ++ AbstractSheaf (missing documentation)
  • AbstractSheaf ++ ZZ (missing documentation)
  • AbstractSheaf - AbstractSheaf (missing documentation)
  • AbstractSheaf ^ ZZ (missing documentation)
  • AbstractSheaf ^** QQ (missing documentation)
  • AbstractSheaf ^** RingElement (missing documentation)
  • AbstractSheaf ^** ZZ (missing documentation)
  • AbstractSheaf QQ (missing documentation)
  • AbstractSheaf RingElement (missing documentation)
  • AbstractSheaf ZZ (missing documentation)
  • adams(ZZ,AbstractSheaf) (missing documentation)
  • ch(AbstractSheaf) (missing documentation)
  • ch(ZZ,AbstractSheaf) (missing documentation)
  • chern(AbstractSheaf) (missing documentation)
  • chern(ZZ,AbstractSheaf) (missing documentation)
  • chern(ZZ,ZZ,AbstractSheaf) -- Get the Chern class of an abstract sheaf
  • chi(AbstractSheaf) (missing documentation)
  • ctop(AbstractSheaf) (missing documentation)
  • determinant(AbstractSheaf) (missing documentation)
  • dual(AbstractSheaf) (missing documentation)
  • End(AbstractSheaf) (missing documentation)
  • exteriorPower(ZZ,AbstractSheaf) (missing documentation)
  • flagBundle(List,AbstractSheaf) (missing documentation)
  • Hom(AbstractSheaf,AbstractSheaf) (missing documentation)
  • net(AbstractSheaf) (missing documentation)
  • projectiveBundle(AbstractSheaf) (missing documentation)
  • rank(AbstractSheaf) (missing documentation)
  • schur(List,AbstractSheaf) (missing documentation)
  • segre(AbstractSheaf) (missing documentation)
  • segre(ZZ,AbstractSheaf) (missing documentation)
  • symmetricPower(QQ,AbstractSheaf) (missing documentation)
  • symmetricPower(RingElement,AbstractSheaf) (missing documentation)
  • symmetricPower(ZZ,AbstractSheaf) (missing documentation)
  • todd(AbstractSheaf) (missing documentation)
  • variety(AbstractSheaf) (missing documentation)
  • ZZ + AbstractSheaf (missing documentation)
  • ZZ ++ AbstractSheaf (missing documentation)

For the programmer

The object AbstractSheaf is a type, with ancestor classes MutableHashTable < HashTable < Thing.