Segre classes and invariants of singular varieties
Paolo Aluffi
TL;DR
This survey demonstrates that Segre classes provide a unifying, algebraic framework for a broad range of invariants associated with singular varieties, including multiplicity, local Euler obstruction, Milnor numbers, various notions of characteristic classes, and Lê cycles. It presents key constructions that express classical invariants as Segre-theoretic data, notably showing how the Chern–Schwartz–MacPherson class can be computed from Segre classes of singularity schemes and cones, and extends this perspective to Milnor and Lê-theoretic invariants. The work emphasizes birational and functorial properties of Segre classes, residual intersection, and projective-cone techniques to enable explicit computations, even in non-smooth settings, with practical implications for algorithmic computation in tools like Macaulay2. Overall, Segre-class methods yield both conceptual clarity and computational pathways for understanding singularities across arbitrary characteristic-zero settings. The results tie together hypersurface theory, polar theory, and modern characteristic-class frameworks into a cohesive algebraic approach to singularities.
Abstract
Segre classes encode essential intersection-theoretic information concerning vector bundles and embeddings of schemes. In this paper we survey a range of applications of Segre classes to the definition and study of invariants of singular spaces. We will focus on several numerical invariants, on different notions of characteristic classes for singular varieties, and on classes of Le cycles. We precede the main discussion with a review of relevant background notions in algebraic geometry and intersection theory.