Generation of jets and Fujita's jet ampleness conjecture on toric varieties

TL;DR

This paper establishes sharp bounds for k-jet ampleness of line bundles on toric varieties, linking geometric properties to combinatorial data, and extends Fujita's conjecture to this setting.

Contribution

It provides explicit criteria for k-jet ampleness on toric varieties and generalizes Fujita's conjecture to varieties with arbitrary singularities.

Findings

Tensor powers of ample line bundles generate all k-jets for certain bounds.

Sharp bounds are given in terms of intersection numbers, lattice lengths, and Seshadri constants.

Proves k-jet versions of Fujita's conjecture on toric varieties.

Abstract

Jet ampleness of line bundles generalizes very ampleness by requiring the existence of enough global sections to separate not just points and tangent vectors, but also their higher order analogues called jets. We give sharp bounds guaranteeing that a line bundle on a projective toric variety is $k$-jet ample in terms of its intersection numbers with the invariant curves, in terms of the lattice lengths of the edges of its polytope, in terms of the higher concavity of its piecewise linear function and in terms of its Seshadri constant. For example, the tensor power $k+n-2$ of an ample line bundle on a projective toric variety of dimension $n \geq 2$ always generates all $k$-jets, but might not generate all $(k+1)$-jets. As an application, we prove the $k$-jet generalizations of Fujita's conjectures on toric varieties with arbitrary singularities.