Betti tables of MCM modules over the cone of a plane cubic
We show that for maximal Cohen–Macaulay modules over the homogeneous coordinate ring of a smooth Calabi–Yau varieties X, the computation of Betti numbers can be reduced to computations of dimensions of certain HomHom spaces in the bounded derived category Db(X). In the simplest case of a smooth elliptic curve E embedded in P2 as a smooth cubic, we get explicit values for Betti numbers. The 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 shifts in internal degree, and two possible shapes up to shifts in internal degree and taking syzygies.