Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces

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.

Some typos corrected and minor changes, citations added to the introduction, references updated