Gauss composition for \(\mathbb{P}^1\), and the universal Jacobian of the Hurwitz space of double covers (Q335612)

From the author's abstract: The questions of how to compactify the Jacobian of a singular curve and how to extend the universal Picard variety over various moduli spaces of curves have been studied extensively in the last several decades.NEWLINENEWLINEIn this paper, we answer those questions for hyperelliptic curves. We give an explicit description of the moduli space of line bundles on hyperelliptic curves, including singular curves. We study the universal Jacobian \(J^{2,g,n}\) of degree \(n\) line bundles over the Hurwitz stack of double covers of \(\mathbb P^{1}\) by a curve of genus \(g.\) Our main results are: the construction of a smooth, irreducible, universally closed (but not separated) moduli compactification \(\overline{J}^{2,g,n}_{bd}\) of \(J^{2,g,n}\) whose points we describe simply and explicitly as sections of certain vector bundles on \(\mathbb P^{1};\) a description of the global geometry and moduli properties of these stacks; and a computation of the Picard groups of \(\overline{J}^{2,g,n}_{bd}\) and \(J^{2,g,n}\) 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.