A-infinity algebras associated with curves and rational functions on M_{g,g}. I

TL;DR

This paper investigates the A-infinity algebra structures on Ext-algebras of sheaves on curves, revealing conditions under which certain higher products vanish or are nontrivial, and explores associated rational maps on moduli spaces.

Contribution

It characterizes the homotopy class of the product m_3 in terms of hyperelliptic curves and Weierstrass points, and determines the structure of the A-infinity algebra for specific cases, linking it to moduli space maps.

Findings

m_3 is homotopic to zero iff the curve is hyperelliptic with Weierstrass points

m_4 is not homotopic to zero for genus ≥ 2 in certain cases

The rational map from moduli space is birational for g≥6 and dominant for g≤5

Abstract

We consider the natural A-infinity structure on the Ext-algebra $Ext^*(G,G)$ associated with the coherent sheaf $G={\cal O}_C\oplus {\cal O}_{p_1}\oplus...\oplus {\cal O}_{p_n}$ on a smooth projective curve $C$, where $p_1,...,p_n\in C$ are distinct points. We study the homotopy class of the product $m_3$. Assuming that $h^0(p_1+...+p_n)=1$ we prove that $m_3$ is homotopic to zero if and only if $C$ is hyperelliptic and the points $p_i$ are Weierstrass points. In the latter case we show that $m_4$ is not homotopic to zero, provided the genus of $C$ is at least 2. In the case $n=g$ we prove that the A-infinity structure is determined uniquely (up to homotopy) by the products $m_i$ with $i\le 6$. Also, in this case we study the rational map ${\cal M}_{g,g}\to {\Bbb A}^{g^2-2g}$ associated with the homotopy class of $m_3$. We prove that for $g\ge 6$ it is birational onto its image, while for $g\le 5$ it is dominant. We also give an interpretation of this map in terms of tangents to $C$ in the canonical embedding and in the projective embedding given by the linear series $|2(p_1+...+p_g)|$.