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MODULI OF CURVES WITH NONSPECIAL DIVISORS AND RELATIVE MODULI OF
$A_{\infty }$-STRUCTURES
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 18 / Issue 6 / November 2019
- Published online by Cambridge University Press:
- 30 October 2017, pp. 1295-1329
- Print publication:
- November 2019
-
- Article
- Export citation
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In this paper, for each
$n\geqslant g\geqslant 0$ we consider the moduli stack 
$\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ of curves 
$(C,p_{1},\ldots ,p_{n},v_{1},\ldots ,v_{n})$ of arithmetic genus 
$g$ with 
$n$ smooth marked points 
$p_{i}$ and nonzero tangent vectors 
$v_{i}$ at them, such that the divisor 
$p_{1}+\cdots +p_{n}$ is nonspecial (has 
$h^{1}=0$) 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 
$\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ with a certain relative moduli of 
$A_{\infty }$-structures (up to an equivalence) over a family of graded associative algebras parametrized by 
$G(n-g,n)$.
