Quadratic Segre indices

TL;DR

This paper establishes a formula for the local Euler class of a line on certain hypersurfaces, linking it to Segre involution indices, and extends the concept over any perfect field of characteristic not 2.

Contribution

It generalizes the concept of Segre involution indices to a broader algebraic setting and connects them to local Euler classes in enumerative geometry.

Findings

Proves the local Euler class equals a product of Segre involution indices.

Extends Segre involution concepts to perfect fields of characteristic not 2.

Provides an infinite family of problems in enriched enumerative geometry.

Abstract

We prove that the local Euler class of a line on a degree $2n-1$ hypersurface in projective $n+1$ space is given by a product of indices of Segre involutions. Segre involutions and their associated indices were first defined by Finashin and Kharlamov over the reals. Our result is valid over any perfect field of characteristic not 2 and gives an infinite family of problems in enriched enumerative geometry with a shared geometric interpretation for the local type.