Newton polyhedra and the integral closure of ideals on toric varieties

TL;DR

This paper extends the understanding of Newton polyhedra and integral closure of ideals to affine toric varieties, characterizing non-degenerate ideals and their closures through geometric and algebraic tools.

Contribution

It generalizes Saia's results to toric varieties, linking integral closure of ideals with Newton polyhedra in this broader context.

Findings

Integral closure of non-degenerate ideals is generated by monomials from the Newton polyhedron.

Characterization of non-degenerate ideals in affine toric varieties.

Extension of classical results to a more general algebraic geometric setting.

Abstract

In this work, we extend Saia's results on the characterization of Newton non-degenerate ideals to the context of ideals in $O_{X(S)}$, where $X(S)$ is an affine toric variety defined by the semigroup $S\subset \mathbb{Z}^{n}_{+}$. We explore the relationship between the integral closure of ideals and the Newton polyhedron. We introduce and characterize non-degenerate ideals, showing that their integral closure is generated by specific monomials related to the Newton polyhedron.