separatesJets -- computes up to which order a toric vector bundle separates jets

Synopsis

Usage:

l = separateJets E

b = isGloballyGenerated E

b = isVeryAmple E
Inputs:
- E, a vector bundle on a toric variety (Klyachko's description)
Outputs:
- l, an integer
- b, a Boolean value
Consequences:
- The result of separatesJets will be stored as a cacheValue in E.cache#separatesJets. It will be used by other methods.

Description

Given a toric vector bundle in Klyachko's description, separatesJets determines up to which order the vector bundle separates jets. Note that a toric vector bundle is globally generated or very ample, if it separates 0-jets or 1-jets, respectively, see [RJS, Theorem 1.2, 6.2 and 6.5]. Hence, the methods isGloballyGenerated and isVeryAmple only ask whether separatesJets returns a non-negative or positive integer, respectively.
If the vector bundle is not even globally generated, then separatesJets returns the value -infinity.
separatesJets calls internally the methods parliament and toricChernCharacter; whereas isGloballyGenerated and isVeryAmple are simple checks on the output of separatesJets.

i1 : E = tangentBundle(projectiveSpaceFan 2)

o1 = {dimension of the variety => 2 }
      number of affine charts => 3
      number of rays => 3
      rank of the vector bundle => 2

o1 : ToricVectorBundleKlyachko

i2 : separatesJets E

o2 = 1

o2 : QQ

i3 : isGloballyGenerated E

o3 = true

i4 : isVeryAmple E

o4 = true

In this example, the vector bundle $\mathcal E$ separates 1-jets, hence is very ample.

Caveat

This methods work for toric vector bundles on a complete simplicial toric variety.
[RJS, Theorem 6.2, condition (iv)] is not checked, which might give a wrong result in special cases (too many polytopes of the parliament share common vertices). If this happens, the method separatesJets will print a warning. In these cases, also the results of isGloballyGenerated and isVeryAmple might be wrong.

Ways to use separatesJets :

separatesJets(ToricVectorBundleKlyachko)

For the programmer

The object separatesJets is a method function with options.