Aspects of Enumerative Geometry with Quadratic Forms

Abstract

Using the motivic stable homotopy category over a field $k$, a smooth variety $X$ over $k$ has an Euler characteristic $\chi (X/k)$ in the Grothendieck-Witt ring $\mathrm{GW}(k)$. The rank of $\chi (X/k)$ is the classical $\mathbb{Z}$-valued Euler characteristic, defined using singular cohomology or étale cohomology, and the signature of $\chi (X/k)$ under a real embedding $\sigma :k\to \mathbb{R}$ gives the topological Euler characteristic of the real points $X^{\sigma }(\mathbb{R})$.

We develop tools to compute $\chi (X/k)$, assuming $k$ has characteristic $\ne 2$ and apply these to refine some classical formulas in enumerative geometry, such as formulas for the top Chern class of the dual, symmetric powers and tensor products of bundles, to identities for the Euler classes in Chow-Witt groups. We also refine the classical Riemann-Hurwitz formula to an identity in $\mathrm{GW}(k)$ and compute $\chi (X/k)$ for hypersurfaces in ${\mathbb{P}}_k^{n+1}$ defined by a polynomial of the form ${\sum }_{i=0}^{n+1}{a}_i{X}_i^m$; this latter includes the case of an arbitrary quadric hypersurface.

This paper is a revision of [M. Levine,"Toward an enumerative geometry with quadratic forms", Preprint, arXiv:1703.03049v3].