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 (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 43c3f230d25c8f4d9f06d7" title="Click to view the MathML source">O_C(p₁+…+p_n)${\mathcal{O}}_C(p_1+\dots +p_n)$ is ample and 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 a_i≥0$a_i\ge 0$ and ∑a_i=g$\sum a_i=g$. We show that each such moduli space ${\tilde{\mathcal{U}}}_{g,n}^{ns}(\mathbf{a})$ 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 ${\tilde{\mathcal{U}}}_{g,n}^{ns}(\mathbf{a})$ by the natural torus action and show that some of the corresponding stack quotients give modular compactifications of 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 43144cd211de7434c2f" title="Click to view the MathML source">n=1$n=1$ corresponds to specifying the Weierstrass gap sequence at the marked point.