How many hypersurfaces does it take to cut out a Segre class?

TL;DR

This paper establishes an identity for Segre classes of zero-schemes, providing bounds on equations needed to define schemes with given Segre classes and a Segre-Bertini theorem for singularity subschemes, with applications in enumerative geometry.

Contribution

It introduces a new identity for Segre classes of zero-schemes and a Segre-Bertini theorem, advancing the understanding of Segre classes in algebraic geometry.

Findings

Derived bounds on the number of equations to define schemes with specific Segre classes

Proved a Segre-Bertini theorem for singularity subschemes of hypersurfaces

Connected Segre class identities to multiplicities and integral closures

Abstract

We prove an identity of Segre classes for zero-schemes of compatible sections of two vector bundles. Applications include bounds on the number of equations needed to cut out a scheme with the same Segre class as a given subscheme of (for example) a projective variety, and a `Segre-Bertini' theorem controlling the behavior of Segre classes of singularity subschemes of hypersurfaces under general hyperplane sections. These results interpolate between an observation of Samuel concerning multiplicities along components of a subscheme and facts concerning the integral closure of corresponding ideals. The Segre-Bertini theorem has applications to characteristic classes of singular varieties. The main results are motivated by the problem of computing Segre classes explicitly and applications of Segre classes to enumerative geometry.