Toric vector bundles and parliaments of polytopes

TL;DR

This paper introduces convex polytopes associated with torus-equivariant vector bundles on toric varieties, linking lattice points to global sections and exploring properties like ampleness and global generation.

Contribution

It presents a novel polytope construction for toric vector bundles, connecting geometric properties to combinatorial data, and provides examples of bundles with specific ampleness properties.

Findings

Lattice points in polytopes correspond to global sections.

Edges of polytopes relate to jets of sections.

Examples of ample bundles that are not globally generated.

Abstract

We introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to generators for the space of global sections and we relate edges to jets. Using the polytopes, we also exhibit toric vector bundles that are ample but not globally generated, and toric vector bundles that are ample and globally generated but not very ample.