A(infinity)-algebras associated with curves and rational functions on M-g,M-g. I

作者：Fisette Robert^*; Polishchuk Alexander

来源：Compositio Mathematica, 2014, 150(4): 621-667.

DOI：10.1112/80010437X13007574

• A(infinity)-structure
• Massey products
• derived category
• Hochschild cohomology
• algebraic curve

We consider the natural A(infinity)-structure on the Ext-algebra Ext*(G, G) associated with the coherent sheaf G = O-C circle plus Op(1) circle plus ... circle plus O-pn on a smooth projective curve C, where p1,..,p(n) is an element of 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 greater than 1. 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 <= 6. Also, in this case we study the rational map M-g,M-g -> A(g2-2g) associated with the homotopy class of m(3). We prove that for g >= 6 it is birational onto its image, while for g <= 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 vertical bar 2(p1 + ... + P-g)vertical bar.

• 出版日期2014-4