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

- 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 (missing documentation) 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

These 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.

## See also

- ToricVectorBundleKlyachko -- the class of all toric vector bundles in Klyachko's description
- parliament -- computes the parliament of polytopes to a toric vector bundle
- compatibleBases -- computes compatible bases for a toric vector bundle
- toricChernCharacter -- computes the toric Chern character of a toric vector bundle

## Ways to use separatesJets:

- `separatesJets(ToricVectorBundleKlyachko)`

## For the programmer

The object separatesJets is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PositivityToricBundles.m2:1115:0.