Stratification of Projection Maps From Toric Varieties

TL;DR

This paper introduces a combinatorial approach to stratify projection maps from toric varieties, providing an algorithm that outperforms existing methods in computing Whitney stratifications.

Contribution

It develops a constructive combinatorial version of Thom's Isotopy Lemma tailored for toric varieties, linking Whitney strata to polytope faces.

Findings

Algorithm outperforms existing methods in examples

Provides a constructive, combinatorial stratification method

Applicable to both complex and real toric varieties

Abstract

We prove a combinatorial version of Thom's Isotopy Lemma for projection maps applied to any complex or real toric variety. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the faces of the polytope of the corresponding toric variety. For all examples we produced, our resulting algorithm outperforms known general purpose methods in Helmer and Nanda (FoCM, 2022), and Dinh and Jelonek (DCG, 2021) for computing map-stratifications.