Computing the Chern-Schwartz-MacPherson Class of Complete Simplical Toric Varieties

TL;DR

This paper presents an effective combinatorial algorithm for computing the Chern-Schwartz-MacPherson class of complete simplicial toric varieties using fan data, implemented in Macaulay2.

Contribution

It introduces a new combinatorial method to compute a specific characteristic class of toric varieties from fan data, enhancing computational tools.

Findings

Algorithm successfully computes Chern-Schwartz-MacPherson classes

Implementation in Macaulay2 demonstrates practical applicability

Method combines and modifies existing theoretical results

Abstract

Topological invariants such as characteristic classes are an important tool to aid in understanding and categorizing the structure and properties of algebraic varieties. In this note we consider the problem of computing a particular characteristic class, the Chern-Schwartz-MacPherson class, of a complete simplicial toric variety X defined by a fan from the combinatorial data contained in the fan. Specifically, we give an effective combinatorial algorithm to compute the Chern-Schwartz-MacPherson class of X, in the Chow ring (or rational Chow ring) of X. This method is formulated by combining, and when necessary modifying, several known results from the literature and is implemented in Macaulay2 for test purposes.