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

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 $|2(p_1+...+p_g)|$.