Desingularization of binomial varieties in arbitrary characteristic. Part II. Combinatorial desingularization algorithm

TL;DR

This paper presents a combinatorial algorithm for resolving singularities of binomial ideals over fields of any characteristic, extending resolution techniques to a broad class of algebraic varieties.

Contribution

It introduces a novel combinatorial resolution algorithm applicable to all binomial ideals, including monomials and p-th powers, without relying on toric geometry tools.

Findings

Algorithm successfully resolves singularities of binomial ideals in arbitrary characteristic.

Applicable to all binomial ideals, including monomials and p-th powers.

Works for toric ideals without using toric geometry tools.

Abstract

In this paper we construct a combinatorial algorithm of resolution of singularities for binomial ideals, over a field of arbitrary characteristic. This algorithm is applied to any binomial ideal. This means ideals generated by binomial equations without any restriction, including monomials and $p$-th powers, where $p$ is the characteristic of the base field. In particular, this algorithm works for toric ideals. However, toric geometry tools are not needed, the algorithm is constructed following the same point of view as Villamayor algorithm of resolution of singularities in characteristic zero.