Moduli of curves with nonspecial divisors and relative moduli of $A_{\infty }$-structures

Abstract

In this paper for each $n\ge g\ge 0$ we consider the moduli stack ${\tilde{\mathcal{U}}}_{g,n}^{ns}$ of curves $(C,p_1,\dots ,p_n,v_1,\dots ,v_n)$ of arithmetic genus $g$ with $n$ smooth marked points $p_i$ and nonzero tangent vectors $v_i$ at them, such that the divisor $p_1+\dots +p_n$ is nonspecial (has no $h^1$) and ample. With some mild restrictions on the characteristic we show that it is a scheme, affine over the Grassmannian $G(n-g,n)$. We also construct an isomorphism of ${\tilde{\mathcal{U}}}_{g,n}^{ns}$ with a certain relative moduli of $A_{\infty }$-structures (up to an equivalence) over a family of graded associative algebras parametrized by $G(n-g,n)$.

v1: 29 pages; v2: 32 pages: added Section 1.3 on the gluing morphism, minor corrections

References in corpus (3)

• Arithmetic mirror symmetry for the 2-torus
• Moduli of curves as moduli of A-infinity structures
• A modular compactification of ${\mathcal{M}}_{1,n}$ from $A_{\infty }$-structures

Cited by in corpus (2)

• A modular compactification of ${\mathcal{M}}_{1,n}$ from $A_{\infty }$-structures
• $A_{\infty }$-structures associated with pairs of 1-spherical objects and noncommutative orders over curves