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

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• 作者：Daniel Erman^a ; ^{erman@umich.edu" class="auth_mail" title="E-mail the corresponding author} ; Melanie Matchett Wood^b ; ^c ; ^{mmwood@math.wisc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Jacobians ; Universal Jacobians ; Hyperelliptic curves ; Compactifications of moduli spaces ; Line bundles on hyperelliptic curves
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：470
• 期：Complete
• 页码：320-352
• 全文大小：690 K

文摘

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 method=retrieve&_eid=1-s2.0-S0021869316302964&_mathId=si2.gif&_user=111111111&_pii=S0021869316302964&_rdoc=1&_issn=00218693&md5=ce544af01164a3c9fa81aacbd826df5e" title="Click to view the MathML source">J^2,g,n${\mathcal{J}}^{2,g,n}$ of degree n   line bundles over the Hurwitz stack of double covers of method=retrieve&_eid=1-s2.0-S0021869316302964&_mathId=si1.gif&_user=111111111&_pii=S0021869316302964&_rdoc=1&_issn=00218693&md5=3cf9872936d1459940a525c885862c4b" title="Click to view the MathML source">P¹${\mathbb{P}}^1$ by a curve of genus g  . Our main results are: the construction of a smooth, irreducible, universally closed (but not separated) moduli compactification method=retrieve&_eid=1-s2.0-S0021869316302964&_mathId=si3.gif&_user=111111111&_pii=S0021869316302964&_rdoc=1&_issn=00218693&md5=021444a43d98fc52a9676fa17163c77e">${\bar{\mathcal{J}}}_{\text{bd}}^{2,g,n}$ of method=retrieve&_eid=1-s2.0-S0021869316302964&_mathId=si2.gif&_user=111111111&_pii=S0021869316302964&_rdoc=1&_issn=00218693&md5=ce544af01164a3c9fa81aacbd826df5e" title="Click to view the MathML source">J^2,g,n${\mathcal{J}}^{2,g,n}$ whose points we describe simply and explicitly as sections of certain vector bundles on method=retrieve&_eid=1-s2.0-S0021869316302964&_mathId=si1.gif&_user=111111111&_pii=S0021869316302964&_rdoc=1&_issn=00218693&md5=3cf9872936d1459940a525c885862c4b" title="Click to view the MathML source">P¹${\mathbb{P}}^1$; a description of the global geometry and moduli properties of these stacks; and a computation of the Picard groups of method=retrieve&_eid=1-s2.0-S0021869316302964&_mathId=si3.gif&_user=111111111&_pii=S0021869316302964&_rdoc=1&_issn=00218693&md5=021444a43d98fc52a9676fa17163c77e">${\bar{\mathcal{J}}}_{\text{bd}}^{2,g,n}$ and method=retrieve&_eid=1-s2.0-S0021869316302964&_mathId=si2.gif&_user=111111111&_pii=S0021869316302964&_rdoc=1&_issn=00218693&md5=ce544af01164a3c9fa81aacbd826df5e" title="Click to view the MathML source">J^2,g,n${\mathcal{J}}^{2,g,n}$ in the cases when method=retrieve&_eid=1-s2.0-S0021869316302964&_mathId=si4.gif&_user=111111111&_pii=S0021869316302964&_rdoc=1&_issn=00218693&md5=9d2a5832e5a3cae8dd02b9ff78062957" title="Click to view the MathML source">n−g$n-g$ 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.