Betti Tables of MCM Modules Over the Cone of a Plane Cubic

TL;DR

This paper develops a method to compute Betti numbers of maximal Cohen-Macaulay modules over the coordinate rings of smooth Calabi-Yau varieties, with explicit results for elliptic curves, using derived category techniques.

Contribution

It introduces a new approach linking Betti number computations to Hom group dimensions in the derived category, providing explicit Betti table classifications for elliptic curves.

Findings

Betti tables have only four possible shapes up to shifts in degree.

Betti tables have only two possible shapes up to shifts and syzygies.

Explicit Betti number formulas are obtained for elliptic curves.

Abstract

We show that for maximal Cohen-Macaulay modules over a homogeneous coordinate rings of smooth Calabi-Yau varieties $X$ computation of Betti numbers can be reduced to computations of dimensions of certain $\operatorname{Hom}$ groups in the bounded derived category $D^b(X)$. In the simplest case of a smooth elliptic curve $E$ imbedded into $\mathbb{P}^2$ as a smooth cubic we use our formula to get explicit answers for Betti numbers. Description of the automorphism group of the derived category $D^b(E)$ 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 a shifts in internal degree, and two possible shapes up to a shift in internal degree and taking syzygies.