$A_{\infty }$-algebras associated with curves and rational functions on $\mathcal{M}_{g, g}$. I. (April 2014)

• Record Type: Journal Article
• Authors: Fisette, Robert; Polishchuk, Alexander
• Is Part Of: Compositio mathematica. Volume 150:Number 4(2014:Jul.)
• Page Start: 621
• Page End: 667
• Publication Date: 2014-04
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X13007574
• Languages: English
• ISSNs: 0010-437X

Abstract

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