Abstract
In this paper we study the moduli stack \({\mathcal {U}}_{1,n}^{ns}\) of curves of arithmetic genus 1 with n marked points, forming a nonspecial divisor. In Polishchuk (A modular compactification of \(\mathcal {M}_{1,n}\) from \(A_\infty \)-structures, arXiv:1408.0611) this stack was realized as the quotient of an explicit scheme \(\widetilde{{\mathcal {U}}}_{1,n}^{ns}\), affine of finite type over \({\mathbb {P}}^{n-1}\), by the action of \({\mathbb {G}}_m^n\) . Our main result is an explicit description of the corresponding GIT semistable loci in \(\widetilde{{\mathcal {U}}}_{1,n}^{ns}\). This allows us to identify some of the GIT quotients with some of the modular compactifications of \({\mathcal {M}}_{1,n}\) defined in Smyth (Invent Math 192:459–503, 2013; Compos Math 147(3):877–913, 2011).
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Supported in part by the NSF Grant DMS-1400390 and by the Russian Academic Excellence Project ‘5-100’.
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Polishchuk, A. Moduli spaces of nonspecial pointed curves of arithmetic genus 1. Math. Ann. 369, 1021–1060 (2017). https://doi.org/10.1007/s00208-017-1562-y
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DOI: https://doi.org/10.1007/s00208-017-1562-y