Quadratic Segre indices
Felipe Espreafico, Stephen McKean, Sabrina Pauli
TL;DR
The paper proves that the local Euler class (local index) of a line on a general degree $2n-1$ hypersurface in $\mathbb{P}^{n+1}$ over a perfect field $k$ with ${\rm char}(k)\neq 2$ is given by a product of Segre indices, unifying and generalizing prior quadratic refinements in real and arithmetic settings. The authors connect the line data to Gauß curves and Castelnuovo secants, introduce conic models to encode Segre data, and show that the local index equals the conic index, which in turn equals the Segre index. They construct two regular maps, $\mathcal R(B,Q)$ and $\mathcal A(B,Q)$, proving the identity $\mathcal A=B,Q) = \mathcal R(B,Q)\cdot (\det\mathmathtt{V}_B)^{2n}$ to relate the conic and local invariants. Consequently, they obtain a quadratically enriched count of lines on $X$ in $\mathrm{GW}(k)$, with real and complex specializations recovering known counts, and provide a broad framework for enriched enumerative geometry of lines on high-degree hypersurfaces. The methods leverage a conic-model perspective to handle non-closed fields and higher dimensions, yielding an infinite family of enriched line-count problems with a shared geometric interpretation for their local indices.
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.