Moduli of curves as moduli of A-infinity structures

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 non-special divisor. We define an explicit closed embedding of a natural ${\mathbb G}_m^g$-torsor over ${\mathcal U}^{ns,a}_{g,g}$ into an affine space and 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 our moduli space of non-special curves is isomorphic to the moduli space of minimal A-infinity structures on a certain finite-dimensional graded associative algebra $E_g$ (introduced in arXiv:1208.6332). We show how to compute explicitly the A-infinity structure associated with a curve $(C,p_1,...,p_g)$ in terms of certain canonical generators of the algebra of functions on $C-\{p_1,...,p_g\}$ and canonical formal parameters at the marked points. We study the GIT quotients associated with our representation of ${\mathcal U}^{ns,a}_{g,g}$ as the quotient of an affine scheme by ${\mathbb G}_m^g$ and show that some of the corresponding stack quotients give modular compactifications of ${\mathcal M}_{g,g}$ in the sense of arXiv:0902.3690. We also consider an analogous picture for curves of arithmetic genus 0 with n marked points which gives a new presentation of the moduli space of $\psi$-stable curves (also known as Boggi-stable curves) and its interpretation in terms of $A_\infty$-structures.