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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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References:
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