Gauss Composition for P^1, and the universal Jacobian of the Hurwitz space of double covers

TL;DR

This paper constructs a compactification of the universal Jacobian over the Hurwitz space of double covers of P^1, describing its geometry, moduli properties, and Picard groups, using a parametrization via binary quadratic forms.

Contribution

It introduces a new smooth, irreducible, universally closed moduli compactification of the universal Jacobian for double covers, generalizing classical quadratic form parametrizations.

Findings

Constructed a smooth, irreducible, universally closed compactification.

Described the global geometry and moduli properties of the stacks.

Computed the Picard groups for cases when n-g is even.

Abstract

We investigate the universal Jacobian of degree n line bundles over the Hurwitz stack of double covers of P^1 by a curve of genus g. Our main results are: the construction of a smooth, irreducible, universally closed (but not separated) moduli compactification of this universal Jacobian; a description of the global geometry and moduli properties of these stacks; and a computation of the Picard groups of these stacks in the cases when n-g is even. An important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms. This parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms, which in particular gives the group law on integral binary quadratic forms first discovered by Gauss.