# 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
• nef, i.e. whether the line bundle $\mathcal{O}(1)$ on $\mathbb{P}(\mathcal E)$ is nef;
• (very) ample, i.e. whether the line bundle $\mathcal{O}(1)$ on $\mathbb{P}(\mathcal E)$ is (very) ample;
• globally generated.
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 developed 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 separate 0-jets or 1-jets, respectively. Here, the central method is separatesJets; built on it are isGloballyGenerated and isVeryAmple.

For the mathematical background see

• [K] Alexander Klyachko, Equivariant bundles over toral varieties, Izv. Akad. Nauk SSSR Ser. Mat., 53, 1989.
• [P] Sam Payne, Moduli of toric vector bundles, Compos. Math, 144, 2008.
• [HMP] Milena Hering, Mircea Mustaţă, Sam Payne, Positivity properties of toric vector bundles, Ann. Inst. Fourier (Grenoble), 60, 2010
• [RJS] Sandra Di Rocco, Kelly Jabbusch, Gregory Smith, Toric vector bundles and parliaments of polytopes, Trans. AMS, 370, 2018.
The following example computes the positivity for the tangent sheaf of \mathbb P^2:
 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 $\mathcal{O}_{\mathbb P^1}(2) \oplus \mathcal{O}_{\mathbb P^1}(1)$.

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
• the fan associated to a polytope will be generated by inner normals,
• the filtrations for describing a toric vector bundle are increasing.
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.

• Polyhedra -- for computations with convex polyhedra, cones, and fans
• ToricVectorBundles -- cohomology computations of equivariant vector bundles on toric varieties

## Author

• Andreas Hochenegger

## 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
• Methods
• "compatibleBases(ToricVectorBundleKlyachko)" -- see compatibleBases -- computes compatible bases for a toric vector bundle
• "drawParliament2Dtikz(ToricVectorBundleKlyachko,String)" -- see drawParliament2Dtikz -- visualises the parliament of polytopes for a vector bundle on a toric surface using TikZ
• "graphToricChernCharacter(ToricVectorBundleKlyachko)" -- see graphToricChernCharacter -- computes the lines connected the components of the toric Chern character of a toric vector bundle
• "groundSet(ToricVectorBundleKlyachko)" -- see groundSet -- computes the ground set of a matroid associated to a toric vector bundle
• "parliament(ToricVectorBundleKlyachko)" -- see parliament -- computes the parliament of polytopes to a toric vector bundle
• "isAmple(ToricVectorBundleKlyachko)" -- see restrictToInvCurves -- computes the restrictions of a toric vector bundle to the torus invariant curves
• "isNef(ToricVectorBundleKlyachko)" -- see restrictToInvCurves -- computes the restrictions of a toric vector bundle to the torus invariant curves
• "restrictToInvCurves(ToricVectorBundleKlyachko)" -- see restrictToInvCurves -- computes the restrictions of a toric vector bundle to the torus invariant curves
• "isGloballyGenerated(ToricVectorBundleKlyachko)" -- see separatesJets -- computes up to which order a toric vector bundle separates jets
• "isVeryAmple(ToricVectorBundleKlyachko)" -- see separatesJets -- computes up to which order a toric vector bundle separates jets
• "separatesJets(ToricVectorBundleKlyachko)" -- see separatesJets -- computes up to which order a toric vector bundle separates jets
• "toricChernCharacter(ToricVectorBundleKlyachko)" -- see 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"

## For the programmer

The object PositivityToricBundles is .