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

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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 Hom groups in the bounded derived category Dᵇ(X). In the simplest case of a smooth elliptic curve E imbedded into P² as a smooth cubic we use our formula to get explicit answers for Betti numbers. Description of the automorphism group of the derived category Dᵇ(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.