Local positivity of line bundles on smooth toric varieties and Cayley polytopes

TL;DR

This paper characterizes smooth toric varieties with maximal local positivity of order k using Cayley polytopes and relates this to the Seshadri constant being exactly k at every point, generalizing previous results.

Contribution

It generalizes earlier characterizations by linking local positivity, Cayley polytopes, and Seshadri constants for smooth toric varieties.

Findings

Maximal k-th osculating dimension corresponds to Cayley polytopes of order k.

Maximal k-th osculating dimension without maximal (k+1)-th relates to Cayley polytopes.

Seshadri constant equals k at every point under these conditions.

Abstract

For any non-negative integer $k$ the $k$-th osculating dimension at a given point $x$ of a variety $X$ embedded in projective space gives a measure of the local positivity of order $k$ at that point. In this paper we show that a smooth toric embedding having maximal $k$-th osculating dimension, but not maximal $(k+1)$-th osculating dimension, at every point is associated to a Cayley polytope of order $k$. This result generalises an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly $k$ at every point of $X$, generalising a result of Atsushi Ito.