Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces

TL;DR

This paper develops methods to compute the quadratic Euler characteristic of smooth hypersurfaces in projective and weighted projective spaces over a field, exploring quadratic refinements of classical formulas and their interpretations.

Contribution

It introduces new techniques for calculating quadratic Euler characteristics and proposes conjectures on quadratic conductor formulas for hypersurface degenerations.

Findings

Derived formulas for quadratic Euler characteristics in specific degenerations.

Established quadratic conductor formulas for hypersurfaces in projective spaces.

Connected quadratic formulas with Ayoub's nearby cycles functor.

Abstract

Let $k$ be a field and let $\text{GW}(k)$ be the Grothendieck-Witt ring of virtual non-degenerate symmetric bilinear forms over $k$. We develop methods for computing the quadratic Euler characteristic $\chi(X/k)\in \text{GW}(k)$ for $X$ a smooth hypersurface in a projective space or a weighted projective space. We raise the question of a quadratic refinement of classical conductor formulas and find such a formula for the degeneration of a smooth hypersurface $X$ in $\mathbb{P}^{n+1}$ to the cone over a smooth hyperplane section of $X$; we also find a similar formula in the weighted homogeneous case. We formulate a conjecture that generalizes these computations to similar types of degenerations. Finally, we give an interpretation of the quadratic conductor formulas in terms of Ayoub's nearby cycles functor.