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

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 $\mathrm{Hom}$ 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.