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MODULI OF CURVES WITH NONSPECIAL DIVISORS AND RELATIVE MODULI OF $A_{\infty }$-STRUCTURES

Part of: Curves Homological methods

Published online by Cambridge University Press:  30 October 2017

Alexander Polishchuk
Alexander Polishchuk*
Affiliation:
University of Oregon, USA (apolish@uoregon.edu) National Research University Higher School of Economics, Russia
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Abstract

In this paper, for each $n\geqslant g\geqslant 0$ we consider the moduli stack $\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ of curves $(C,p_{1},\ldots ,p_{n},v_{1},\ldots ,v_{n})$ of arithmetic genus $g$ with $n$ smooth marked points $p_{i}$ and nonzero tangent vectors $v_{i}$ at them, such that the divisor $p_{1}+\cdots +p_{n}$ is nonspecial (has $h^{1}=0$) and ample. With some mild restrictions on the characteristic we show that it is a scheme, affine over the Grassmannian $G(n-g,n)$. We also construct an isomorphism of $\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ with a certain relative moduli of $A_{\infty }$-structures (up to an equivalence) over a family of graded associative algebras parametrized by $G(n-g,n)$.

Keywords

A∞-algebramoduli of curvesSato Grassmannian

MSC classification

Primary: 14H10: Families, moduli (algebraic)
Secondary: 16E45: Differential graded algebras and applications
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 6 , November 2019 , pp. 1295 - 1329
Copyright
© Cambridge University Press 2017 

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Footnotes

Supported in part by the NSF grant DMS-1400390 and by the Russian Academic Excellence Project ‘5-100’.

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