A Direct Algorithm to Compute the Topological Euler Characteristic and Chern-Schwartz-MacPherson Class of Projective Complete Intersection Varieties

TL;DR

This paper introduces a new algorithm that efficiently computes the topological Euler characteristic and Chern-Schwartz-MacPherson class of projective complete intersection varieties, improving performance over existing methods.

Contribution

It develops a novel algorithm based on recent theoretical results to compute these invariants for singular complete intersections in projective space.

Findings

Algorithm improves computational efficiency for certain complete intersections.

Provides a practical tool for calculating topological invariants of singular varieties.

Enhances existing computational methods with performance gains.

Abstract

Let $V$ be a possibly singular scheme-theoretic complete intersection subscheme of $\mathbb{P}^n$ over an algebraically closed field of characteristic zero. Using a recent result of Fullwood ("On Milnor classes via invariants of singular subschemes", Journal of Singularities) we develop an algorithm to compute the Chern-Schwartz-MacPherson class and Euler characteristic of $V$. This algorithm complements existing algorithms by providing performance improvements in the computation of the Chern-Schwartz-MacPherson class and Euler characteristic for certain types of complete intersection subschemes of $\mathbb{P}^n$.