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Betti numbers of MCM modules over the cone of an elliptic normal curve. (English) Zbl 07040790
Summary: 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 \(\mathbb{P}^{n - 1}\), by the full linear system \(|\mathcal{O}(n x)|\), 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 \(\widetilde{E_8}\) for \(n = 1\), \(\widetilde{E_7}\) for \(n = 2\) and \(\widetilde{E_6}\) for the cone over a smooth cubic, that is, for the case \(n = 3\). For the singularities \(\widetilde{E_7}\) and \(\widetilde{E_8}\) we obtain formulas for the Betti numbers and the numerical invariants of MCM modules analogous to the case of a plane cubic.
MSC:
14H52 Elliptic curves
13C14 Cohen-Macaulay modules
13D02 Syzygies, resolutions, complexes and commutative rings
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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