Macaulay2 » Documentation
Packages » PositivityToricBundles :: compatibleBases

Synopsis

• Usage:

  b = compatibleBases E

• Inputs:
  • E, a vector bundle on a toric variety (Klyachko's description)
• Outputs:
  • b, a hash table, whose keys are maximal cones and the value of a key is a matrix
• Consequences:

  • The result of compatibleBases will be stored as a cacheValue in E.cache#compatibleBases. It will be used by other methods.

Description

Given a toric vector bundle in Klyachko's description, compatibleBases computes for each maximal cone of the underlying toric variety a compatible basis of the filtrations, see [RJS, Section 3] for details.
compatibleBases calls internally the method groundSet.

The compatible bases serve as input for the computation of the toric Chern character by toricChernCharacter.

Caveat

This method works for any toric reflexive sheaf (see isLocallyWeil for an example where the sheaf is not locally free) on a toric variety, whose fan is covered by cones of maximal dimension.

See also

• ToricVectorBundleKlyachko -- the class of all toric vector bundles in Klyachko's description
• groundSet -- computes the ground set of a matroid associated to a toric vector bundle
• toricChernCharacter -- computes the toric Chern character of a toric vector bundle