On $k$-normality and Regularity of Normal Toric Varieties

TL;DR

This paper establishes a new combinatorial bound linking the $k$-normality of very ample lattice polytopes to the Castelnuovo-Mumford regularity of normal projective toric varieties, enhancing understanding of their algebraic and geometric properties.

Contribution

It introduces a novel combinatorial bound for $k$-normality and regularity of normal toric varieties, connecting polytope properties with algebraic invariants.

Findings

Bound of $k$ for $k$-normality of very ample lattice polytopes

New combinatorial bound for Castelnuovo-Mumford regularity

Improved understanding of the relationship between polytopes and toric variety regularity

Abstract

We give a bound of $k$ for a very ample lattice polytope to be $k$-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity of normal projective toric varieties.