## Duke Mathematical Journal

### Moduli of curves as moduli of $A_{\infty}$-structures

Alexander Polishchuk

#### Abstract

We define and study the stack $\mathcal{U}^{ns,a}_{g,g}$ of (possibly singular) projective curves of arithmetic genus $g$ with $g$ smooth marked points forming an ample nonspecial divisor. We define an explicit closed embedding of a natural $\mathbb{G}_{m}^{g}$-torsor $\widetilde{\mathcal{U}}^{ns,a}_{g,g}$ over $\mathcal{U}^{ns,a}_{g,g}$ into an affine space, and we give explicit equations of the universal curve (away from characteristics $2$ and $3$). This construction can be viewed as a generalization of the Weierstrass cubic and the $j$-invariant of an elliptic curve to the case $g\gt 1$. Our main result is that in characteristics different from $2$ and $3$ the moduli space $\widetilde{\mathcal{U}}^{ns,a}_{g,g}$ is isomorphic to the moduli space of minimal $A_{\infty}$-structures on a certain finite-dimensional graded associative algebra $E_{g}$ (introduced by Fisette and Polishchuk).

#### Article information

Source
Duke Math. J., Volume 166, Number 15 (2017), 2871-2924.

Dates
Received: 2 September 2015
Revised: 20 March 2017
First available in Project Euclid: 8 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1504836224

Digital Object Identifier
doi:10.1215/00127094-2017-0019

Mathematical Reviews number (MathSciNet)
MR3712167

Zentralblatt MATH identifier
06812211

#### Citation

Polishchuk, Alexander. Moduli of curves as moduli of $A_{\infty}$ -structures. Duke Math. J. 166 (2017), no. 15, 2871--2924. doi:10.1215/00127094-2017-0019. https://projecteuclid.org/euclid.dmj/1504836224

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