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Moduli of curves as moduli of A_infty-structures
We define and study the stack 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 𝔾gm-torsor over 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 Eg (introduced in arXiv:1208.6332). We show how to compute explicitly the A-infinity structure associated with a curve (C,p1,...,pg) in terms of certain canonical generators of the algebra of functions on C−{p1,...,pg} and canonical formal parameters at the marked points. We study the GIT quotients associated with our representation of U^{ns,a}_{g,g} as the quotient of an affine scheme by 𝔾m^g and show that some of the corresponding stack quotients give modular compactifications of 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 ψ-stable curves (also known as Boggi-stable curves) and its interpretation in terms of A∞-structures.