Duke Mathematical Journal

Moduli of curves as moduli of A-structures

Alexander Polishchuk

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We define and study the stack Ug,gns,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 Gmg-torsor U˜g,gns,a over Ug,gns,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 U˜g,gns,a is isomorphic to the moduli space of minimal A-structures on a certain finite-dimensional graded associative algebra Eg (introduced by Fisette and Polishchuk).

Article information

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

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

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Zentralblatt MATH identifier

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

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


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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