Computing Segre classes in arbitrary projective varieties

TL;DR

This paper introduces a novel algorithm for computing Segre classes of subschemes within any projective variety, including those with singularities, by leveraging degrees of linear projections and intersection properties.

Contribution

The paper presents the first general algorithm capable of computing Segre classes in arbitrary projective varieties, extending beyond previous methods limited to smooth or special cases.

Findings

Algorithm successfully computes Segre classes in varieties with arbitrary singularities.

Uses degrees of linear projections and intersection theory to determine Segre class coefficients.

First known method applicable to general projective varieties with complex singularities.

Abstract

We give an algorithm for computing Segre classes of subschemes of arbitrary projective varieties by computing degrees of a sequence of linear projections. Based on the fact that Segre classes of projective varieties commute with intersections by general effective Cartier divisors, we can compile a system of linear equations which determine the coefficients for the Segre class pushed forward to projective space. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space; to our knowledge, this is the first algorithm to be able to compute Segre classes in projective varieties with arbitrary singularities.