Algorithms to compute the topological Euler characteristic, Chern–Schwartz–MacPherson class and Segre class of projective varieties

Abstract

Let V be a closed subscheme of a projective space Pn. We give an algorithm to compute the Chern–Schwartz–MacPherson class, and the Euler characteristic of V and an algorithm to compute the Segre class of V. The algorithms can be implemented using either symbolic or numerical methods. The algorithms are based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Relationships between the algorithms developed here and other existing algorithms are discussed. The algorithms are tested on several examples and are found to perform favourably compared to current algorithms for computing Chern–Schwartz–MacPherson classes, Segre classes and Euler characteristics.