Moduli of curves as moduli of A-infinity 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 non-special divisor. We define an explicit closed embedding of a natural ${\mathbb{G}}_m^g$-torsor over ${\mathcal{U}}_{g,g}^{ns,a}$ 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 ${\mathcal{U}}_{g,g}^{ns,a}$ 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.

v1: 58 pages; v2: 69 pages; added a section on curves of arithmetic genus 0 and a section on the connection to Petri's analysis of a non-hyperelliptic curve in the canonical embeding; v3: 71 pages; minor improvements; v4: 72 pages, improved presentation in Section 2.4; v5: 74 pages, corrections in the proof of Theorem 1.2.4; v6: filled a gap in the proof of Corollary 4.2.5; minor corrections

References in corpus (1)

• Deformation theory via differential graded Lie algebras

Cited by in corpus (6)

• Homological mirror symmetry for invertible polynomials in two variables
• A modular compactification of ${\mathcal{M}}_{1,n}$ from $A_{\infty }$-structures
• Moduli of curves with nonspecial divisors and relative moduli of $A_{\infty }$-structures
• $A_{\infty }$-structures associated with pairs of 1-spherical objects and noncommutative orders over curves
• A-infinity algebras associated with elliptic curves and Eisenstein-Kronecker series
• Birational models of ${\mathcal{M}}_{2,2}$ arising as moduli of curves with nonspecial divisors