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PositivityToricBundles :: PositivityToricBundles

PositivityToricBundles -- checks positivity of toric vector bundles

Description

Given a toric vector bundle, i.e. an equivariant vector bundle on a smooth complete toric variety, this package can check the positivity of this bundle. PositivityToricBundles can check whether a toric vector bundle is Additionally, PositivityToricBundles can compute the toric Chern character of a toric vector bundle as introduced by Sam Payne.

For the computational purposes, PositivityToricBundles uses the description of a toric vector bundles by filtrations developped by Alexander Klyachko, and relies on its implementation via the ToricVectorBundles package by René Birkner, Nathan Ilten and Lars Petersen.
To check nefness and ampleness, PositivityToricBundles uses a result of Milena Hering, Mircea Mustaţă and Sam Payne, namely, that it is sufficient to check this for the restriction of the bundle to the torus invariant curves. The central method for this is restrictToInvCurves; the methods isNef and isAmple are based on it.
For global generation and very ampleness, PositivityToricBundles uses results of Sandra Di Rocco, Kelly Jabbusch and Gregory Smith, who describe these properties in terms of the so-called parliament of polytopes of a toric vector bundle. From the parliament of polytopes one can extract the information up to which order jets are separated by the vector bundle. Globally generated or very ample toric vector bundles are those that separete 0-jets or 1-jets, respectively. Here, the central method is separatesJets; built on it are isGloballyGenerated and isVeryAmple.

For the mathematical background see

The following example computes the positivity for the tangent sheaf of ℙ2:
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 : isNef E

o2 = true
i3 : isAmple E

o3 = true
i4 : isVeryAmple E

o4 = true
i5 : isGloballyGenerated E

o5 = true
i6 : separatesJets E

o6 = 1

The toric Chern character can be computed:

i7 : toricChernCharacter E

o7 = HashTable{| -1 0 | => {| 1 |, | 1  |}}
               | -1 1 |     | 0 |  | -1 |
               | 1 -1 | => {| -1 |, | 0 |}
               | 0 -1 |     | 1  |  | 1 |
               | 1 0 | => {| -1 |, | 0  |}
               | 0 1 |     | 0  |  | -1 |

o7 : HashTable
which associates to each maximal cone (its rays put into matrices) the corresponding components.

The restrictions of the bundle to the torus invariant curves can be computed:

i8 : restrictToInvCurves E

o8 = HashTable{| -1 | => {2, 1}}
               | -1 |
               | 0 | => {2, 1}
               | 1 |
               | 1 | => {1, 2}
               | 0 |

o8 : HashTable
Here, in all three cases, the restriction splits into O1(2) ⊕O1(1).

Most methods of PositivityToricBundles support the option Verbosity. So by adding Verbosity => n with n a positive integer to the arguments of a method, hopefully useful insight about the course of the calculation is provided.

Caveat

The description of a toric variety and a toric vector bundle by filtrations involves the choice of signs. PositivityToricBundles follows the same choice of signs as ToricVectorBundles, which are Unfortunately, the above cited articles use decreasing filtrations and, moreover, [HMP], [P] and [RJS] use outer normals.

Another warning concerns the toric variety: the methods of PositivityToricBundles implicitly assume that the variety is complete (to apply the results of [HMP] and [P]) and in addition smooth (for [RJS]). For non-complete or singular toric varieties, methods might break or results might become meaningless.

See also

Author

Version

This documentation describes version 1.1 of PositivityToricBundles.

Source code

The source code from which this documentation is derived is in the file PositivityToricBundles.m2.

Exports

  • Functions and commands
    • compatibleBases -- computes compatible bases for a toric vector bundle
    • drawParliament2Dtikz -- visualises the parliament of polytopes for a vector bundle on a toric surface using TikZ
    • graphToricChernCharacter -- computes the lines connected the components of the toric Chern character of a toric vector bundle
    • groundSet -- computes the ground set of a matroid associated to a toric vector bundle
    • parliament -- computes the parliament of polytopes to a toric vector bundle
    • isAmple, see restrictToInvCurves -- computes the restrictions of a toric vector bundle to the torus invariant curves
    • isNef, see restrictToInvCurves -- computes the restrictions of a toric vector bundle to the torus invariant curves
    • restrictToInvCurves -- computes the restrictions of a toric vector bundle to the torus invariant curves
    • isGloballyGenerated, see separatesJets -- computes up to which order a toric vector bundle separates jets
    • separatesJets -- computes up to which order a toric vector bundle separates jets
    • toricChernCharacter -- computes the toric Chern character of a toric vector bundle
  • Symbols
    • DrawChernCharacter, see drawParliament2Dtikz -- visualises the parliament of polytopes for a vector bundle on a toric surface using TikZ
    • DrawCohomology, see drawParliament2Dtikz -- visualises the parliament of polytopes for a vector bundle on a toric surface using TikZ
    • Verbosity