Newton polyhedra and the integral closure of ideals on toric varieties

TL;DR

This work generalizes the theory of Newton non-degenerate ideals to affine toric varieties $X(S)$ and links integral closure to geometric data encoded by Newton polyhedra. It proves that an ideal $I$ is non-degenerate if and only if the Newton polyhedron $\Gamma_{+}(I)$ coincides with $C(\overline{I})$, the polyhedron generated by monomials in the integral closure, and consequently $\overline{I}=I^{\circ}$. A toroidal-embedding construction is developed to realize this criterion and relates it to a concrete analysis of equisingularity via Whitney conditions, yielding practical criteria and examples. The framework provides a combinatorial approach to understanding integral closures on toric varieties, with potential applications to stratification and equisingularity in families of toric or torus-invariant spaces.

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.

Figures (4)

• Figure 1: Newton polyhedron.
• Figure 2: Newton polyhedron of $I=\langle x^2y+3x^2z,y^3-xy+z^2\rangle$.
• Figure 3: Newton polyhedron of $I$.
• Figure 4: Newton polyhedron of $I_F$.

Theorems & Definitions (36)

• Theorem
• Theorem
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Lemma 1.4
• Definition 1.5
• Proposition 2.1
• Theorem 2.2
• proof