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arXiv · 1511.05089

Betti Tables of MCM Modules Over the Cone of a Plane Cubic

Alexander Pavlov
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Abstract

We show that for maximal Cohen-Macaulay modules over a homogeneous coordinate rings of smooth Calabi-Yau varieties $X$ computation of Betti numbers can be reduced to computations of dimensions of certain $\operatorname{Hom}$ groups in the bounded derived category $D^b(X)$. In the simplest case of a smooth elliptic curve $E$ imbedded into $\mathbb{P}^2$ as a smooth cubic we use our formula to get explicit answers for Betti numbers. Description of the automorphism group of the derived category $D^b(E)$ in terms of the spherical twist functors of Seidel and Thomas plays a major role in our approach. We show that there are only four possible shapes of the Betti tables up to a shifts in internal degree, and two possible shapes up to a shift in internal degree and taking syzygies.

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Alexander Pavlov. 2015-11-16. Betti Tables of MCM Modules Over the Cone of a Plane Cubic. https://arxiv.org/abs/1511.05089

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