Castelnuovo polytopes

TL;DR

This paper characterizes Castelnuovo lattice polytopes, extending Kawaguchi's work, and applies this to provide a criterion for lattice polytopes to be IDP.

Contribution

It generalizes Kawaguchi's characterization of Castelnuovo polytopes and introduces a new criterion for lattice polytopes to be integrally closed.

Findings

Complete characterization of Castelnuovo polytopes.

New sufficient criterion for lattice polytopes to be IDP.

Abstract

It is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound on the genus of a projective curve. Polarized varieties whose sectional genus achieves this bound are called Castelnuovo. On the other hand, a lattice polytope is called Castelnuovo if the associated polarized toric variety is Castelnuovo. Kawaguchi characterized Castelnuovo polytopes having interior lattice points in terms of their $h^*$-vectors. In this paper, as a generalization of this result, a characterization of all Castelnuovo polytopes will be presented. Finally, as an application of our characterization, we give a sufficient criterion for a lattice polytope to be IDP.