Betti numbers of MCM modules over the cone of an elliptic normal curve

Abstract

We apply Orlov's equivalence to derive formulas for the Betti numbers of maximal Cohen–Macaulay modules over the cone an elliptic curve (E,x) embedded into Pn−1, by the full linear system |O(nx)|, for n>3. The answers are given in terms of recursive sequences. These results are applied to give a criterion of (Co-)Koszulity.In the last two sections of the paper we apply our methods to study the cases n=1,2. Geometrically these correspond to the embedding of an elliptic curve into a weighted projective space. The singularities of the corresponding cones are called minimal elliptic. They were studied by K. Saito [13], where he introduced the notation E8˜ for n=1, E7˜ for n=2 and E6˜ for the cone over a smooth cubic, that is, for the case n=3. For the singularities E7˜ and E8˜ we obtain formulas for the Betti numbers and the numerical invariants of MCM modules analogous to the case of a plane cubic.