next | previous | forward | backward | up | index | toc

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

## Synopsis

• Usage:
l = separateJets E
b = isGloballyGenerated E
b = isVeryAmple E
• Inputs:
• Outputs:
• 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.