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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_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,nclass="mathContainer hidden">class="mathCode">${\mathcal{J}}^{2,g,n}$ of degree n   line bundles over the Hurwitz stack of double covers of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_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">P¹class="mathContainer hidden">class="mathCode">${\mathbb{P}}^1$ by a curve of genus g  . Our main results are: the construction of a smooth, irreducible, universally closed (but not separated) moduli compactification class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302964&_mathId=si3.gif&_user=111111111&_pii=S0021869316302964&_rdoc=1&_issn=00218693&md5=021444a43d98fc52a9676fa17163c77e">class="imgLazyJSB inlineImage" height="24" width="51" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302964-si3.gif">class="mathContainer hidden">class="mathCode">${\bar{\mathcal{J}}}_{\text{bd}}^{2,g,n}$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_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,nclass="mathContainer hidden">class="mathCode">${\mathcal{J}}^{2,g,n}$ whose points we describe simply and explicitly as sections of certain vector bundles on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_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">P¹class="mathContainer hidden">class="mathCode">${\mathbb{P}}^1$; a description of the global geometry and moduli properties of these stacks; and a computation of the Picard groups of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302964&_mathId=si3.gif&_user=111111111&_pii=S0021869316302964&_rdoc=1&_issn=00218693&md5=021444a43d98fc52a9676fa17163c77e">class="imgLazyJSB inlineImage" height="24" width="51" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302964-si3.gif">class="mathContainer hidden">class="mathCode">${\bar{\mathcal{J}}}_{\text{bd}}^{2,g,n}$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_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,nclass="mathContainer hidden">class="mathCode">${\mathcal{J}}^{2,g,n}$ in the cases when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_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−gclass="mathContainer hidden">class="mathCode">$n-g$ 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.