$A_{∞ }$-algebras associated with curves and rational functions on $\mathcal{M}_{g, g}$. I. (April 2014) - Fisette, Robert; Polishchuk, Alexander
Abstract We consider the natural <alternatives><inline-graphic xlink:href="ark:/27927/pghk62p7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$A_{\infty }$]]> </alternatives>-structure on the <alternatives><inline-graphic xlink:href="ark:/27927/pghk62mb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathrm{Ext}$]]> </alternatives>-algebra <alternatives><inline-graphic xlink:href="ark:/27927/pghk625h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathrm{Ext}^*(G, G)$]]> </alternatives> associated with the coherent sheaf <alternatives><inline-graphic xlink:href="ark:/27927/pghk62gj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$]]> </alternatives> on a smooth projective curve <alternatives><inline-graphic xlink:href="ark:/27927/pghk6420" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$C$]]> </alternatives>, where <alternatives><inline-graphic xlink:href="ark:/27927/pghk63pn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$p_1, \ldots, p_n\in C$]]> </alternatives> are distinct points. We study the homotopy class of the product <alternatives><inline-graphic xlink:href="ark:/27927/pghk63k9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$m_3$]]> </alternatives>. Assuming that <alternatives><inline-graphic xlink:href="ark:/27927/pghk61g4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$h^0(p_1+\cdots +p_n)=1$]]> </alternatives>, we prove that <alternatives><inline-graphic xlink:href="ark:/27927/pghk62sk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$m_3$]]> </alternatives> is homotopic to zero if and only if <alternatives><inline-graphic xlink:href="ark:/27927/pghk62xc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$C$]]> </alternatives> is hyperelliptic and the points <alternatives><inline-graphic xlink:href="ark:/27927/pghk63gz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$p_i$]]> </alternatives> are Weierstrass points. In the latter case we show that <alternatives><inline-graphic xlink:href="ark:/27927/pghk641j" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$m_4$]]> </alternatives> is not homotopic to zero, provided the genus of <alternatives><inline-graphic xlink:href="ark:/27927/pghk6208" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$C$]]> </alternatives> is greater than <alternatives><inline-graphic xlink:href="ark:/27927/pghk61j1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$1$]]> </alternatives>. In the case <alternatives><inline-graphic xlink:href="ark:/27927/pghk61tm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$n=g$]]> </alternatives> we prove that the <alternatives><inline-graphic xlink:href="ark:/27927/pghk61rq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$A_{\infty }$]]> </alternatives>-structure is determined uniquely (up to homotopy) by the products <alternatives><inline-graphic xlink:href="ark:/27927/pghk61wh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$m_i$]]> </alternatives> with <alternatives><inline-graphic xlink:href="ark:/27927/pghk61v2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$i\le 6$]]> </alternatives>. Also, in this case we study the rational map <alternatives><inline-graphic xlink:href="ark:/27927/pghk62r4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathcal{M}_{g, g}\to \mathbb{A}^{g^2-2g}$]]> </alternatives> associated with the homotopy class of <alternatives><inline-graphic xlink:href="ark:/27927/pghk6225" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$m_3$]]> </alternatives>. We prove that for <alternatives><inline-graphic xlink:href="ark:/27927/pghk63jv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$g\ge 6$]]> </alternatives> it is birational onto its image, while for <alternatives><inline-graphic xlink:href="ark:/27927/pghk61cs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$g\le 5$]]> </alternatives> it is dominant. We also give an interpretation of this map in terms of tangents to <alternatives><inline-graphic xlink:href="ark:/27927/pghk636c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$C$]]> </alternatives> in the canonical embedding and in the projective embedding given by the linear series <alternatives><inline-graphic xlink:href="ark:/27927/pghk61fp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$|2(p_1+\cdots +p_g)|$]]> </alternatives>.