Maximal linear spaces contained in the base loci of pencils of quadrics

TL;DR

This paper explores the geometry of maximal linear spaces in the base locus of quadrics over arbitrary fields, linking Fano schemes to Jacobians of hyperelliptic curves and analyzing their structure.

Contribution

It extends the understanding of Fano schemes of quadrics to arbitrary fields and relates them to hyperelliptic curve Jacobians, including cases with singular discriminant curves.

Findings

Fano schemes form components of a disconnected algebraic group related to Jacobians.

The structure of these schemes varies with the singularity of the hyperelliptic curve.

Connections established between geometric configurations and algebraic groups.

Abstract

The geometry of the Fano scheme of maximal linear spaces contained in the base locus of a pencil of quadrics has been studied by algebraic geometers when the base field is algebraically closed. In this paper, we work over an arbitrary base field of characteristic not equal to 2 and show how these Fano schemes are related to the Jacobians of hyperelliptic curves. In particular, if $B$ is the base locus of a generic pencil of quadrics in $\bbp^{2n+1}$, and $F$ is the Fano variety of $n - 1$ planes contained in $B$, then $F$ is a component of a disconnected commutative algebraic group $G = \picz(C) \dcup F \dcup \pico(C) \dcup F'$, where $C$ is the hyperelliptic curve defined by the discriminant form of the pencil. In the second half of this paper, we study regular pencils of quadrics, where the hyperelliptic curve defined by the discriminant is singular.