Moduli of curves, Gröbner bases, and the Krichever map

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• 作者：Alexander Polishchuk¹ ; ^{apolish@uoregon.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Moduli spaces of curves ; Groebner bases ; Krichever map ; GIT quotient ; Sato Grassmannian ; Weierstrass gap sequence
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：682-756
• 全文大小：1198 K

文摘

We study moduli spaces of (possibly non-nodal) curves 8be60be47e360f7" title="Click to view the MathML source">(C,p₁,…,p_n)$(C,p_1,\dots ,p_n)$ of arithmetic genus g with n   smooth marked points, equipped with nonzero tangent vectors, such that e8ff87843c3f230d25c8f4d9f06d7" title="Click to view the MathML source">O_C(p₁+…+p_n)${\mathcal{O}}_C(p_1+\dots +p_n)$ is ample and 99d9190492c9633f3a25fd33" title="Click to view the MathML source">H¹(O_C(a₁p₁+…+a_np_n))=0$H^1({\mathcal{O}}_C(a_1{p}_1+\dots +a_n{p}_n))=0$ for given integer weights a=(a₁,…,a_n)$\mathbf{a}=(a_1,\dots ,a_n)$ such that 9f41d565f97a7b83ab48ec6b3b" title="Click to view the MathML source">a_i≥0$a_i\ge 0$ and 8bea1ed0e" title="Click to view the MathML source">∑a_i=g$\sum a_i=g$. We show that each such moduli space 8b9a84122ba9669">${\tilde{\mathcal{U}}}_{g,n}^{ns}(\mathbf{a})$ is an affine scheme of finite type, and the Krichever map identifies it with the quotient of an explicit locally closed subscheme of the Sato Grassmannian by the free action of the group of changes of formal parameters. We study the GIT quotients of 8b9a84122ba9669">${\tilde{\mathcal{U}}}_{g,n}^{ns}(\mathbf{a})$ by the natural torus action and show that some of the corresponding stack quotients give modular compactifications of e678" title="Click to view the MathML source">M_g,n${\mathcal{M}}_{g,n}$ with projective coarse moduli spaces. More generally, using similar techniques, we construct moduli spaces of curves with chains of divisors supported at marked points, with prescribed number of sections, which in the case e43144cd211de7434c2f" title="Click to view the MathML source">n=1$n=1$ corresponds to specifying the Weierstrass gap sequence at the marked point.