## Duke Mathematical Journal

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

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

#### Abstract

We define and study the stack ${\mathcal{U}}_{g,g}^{ns,a}$ 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 ${\tilde{\mathcal{U}}}_{g,g}^{ns,a}$ over ${\mathcal{U}}_{g,g}^{ns,a}$ 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 $g>1$. Our main result is that in characteristics different from $2$ and $3$ the moduli space ${\tilde{\mathcal{U}}}_{g,g}^{ns,a}$ 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

**Subjects**

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]

Secondary: 14H10: Families, moduli (algebraic) 16E45: Differential graded algebras and applications

**Keywords**

moduli space of curves A-infinity-algebra Hochschild cohomology deformation theory

#### 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