PositivityToricBundles

Macaulay2 package PositivityToricBundles - checks positivity of toric vector bundles. 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. 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.

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