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

TL;DR

This paper uses Orlov's equivalence to derive formulas for Betti numbers of maximal Cohen-Macaulay modules over cones of elliptic curves embedded in projective space, with applications to Koszulity and singularity analysis.

Contribution

It provides explicit recursive formulas for Betti numbers of MCM modules over cones of elliptic curves, extending to singular cases and relating to minimal elliptic singularities.

Findings

Formulas for Betti numbers in terms of recursive sequences

Criteria for (Co-)Koszulity of modules

Explicit results for singular cases $ ilde{E}_7$ and $ ilde{E}_8$

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 $\mathbb{P}^{n-1}$, by the full linear system $|\mathcal{O}(nx)|$, 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 cases 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, 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.