Torical Modification of Newton non-degenerate ideals

TL;DR

This paper extends the concept of Newton non-degeneracy to a broader class of algebraic varieties, providing a toric modification approach for their resolution using tropical and toric geometry tools.

Contribution

It introduces a generator-independent definition of Newton non-degeneracy applicable to non-complete intersection varieties and demonstrates their resolution via toric modifications.

Findings

Varieties satisfying the new non-degeneracy can be resolved torically.

The toric modification is described using the Groebner fan.

The approach generalizes previous hypersurface results.

Abstract

We give a definition of Newton non degeneracy independent of the system of generators defining the variety. This definition extends the notion of Newton non degeneracy to varieties that are not necessarily complete intersection. As in the previous definition of non-degeneracy for complete intersection varieties, it is shown that the varieties satisfying our definition can be resolved with a toric modification. Using tools of both toric and tropical geometry we describe the toric modification in terms of the Groebner fan of the ideal defining the variety. The first part of the paper is devoted to introducing the classical concepts and the proof for the hypersurface case.