Groebner fan and embedded resolutions of ideals on toric varieties

TL;DR

This paper extends the concepts of Groebner fan and Newton non-degeneracy to ideals on toric varieties, proving the fan's polyhedral structure and existence of toric embedded resolutions without characteristic restrictions.

Contribution

It introduces generalized notions of Groebner fan and Newton non-degeneracy for toric varieties and proves their key properties and resolution existence.

Findings

Groebner fan of an ideal on a toric variety is polyhedral.

Newton non-degenerate ideals admit toric embedded resolutions.

Results hold over arbitrary characteristic fields.

Abstract

We consider the notions of Groebner fan and Newton non-degeneracy for an ideal on a toric variety, extending the two existing notions for ideals on affine spaces. We prove, without assumptions on the characteristic of the base fields, that the "Groebner fan" of such an ideal is actually a polyhedral fan and that a sub-variety defined by a Newton non-degenerate ideal on a toric variety $X_\sigma$ admits a toric embedded resolution of singularities $Z\longrightarrow X_\sigma.$