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
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 |
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 |
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.
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.
This documentation describes version 1.1 of PositivityToricBundles.
The source code from which this documentation is derived is in the file PositivityToricBundles.m2.
The object PositivityToricBundles is a package.