Gauss composition for P1P1, and the universal Jacobian of the Hurwitz space of double covers

In this paper, we give an explicit description of the moduli space of line bundles on hyperelliptic curves, including singular curves. We study the universal Jacobian J2,g,nJ2,g,n of degree n   line bundles over the Hurwitz stack of double covers of P1P1 by a curve of genus g  . Our main results are: the construction of a smooth, irreducible, universally closed (but not separated) moduli compactification J‾bd2,g,n of J2,g,nJ2,g,n whose points we describe simply and explicitly as sections of certain vector bundles on P1P1; a description of the global geometry and moduli properties of these stacks; and a computation of the Picard groups of J‾bd2,g,n and J2,g,nJ2,g,n in the cases when n−gn−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.