Betti tables of MCM modules over the cone of a plane cubic

Abstract

We show that for maximal Cohen–Macaulay modules over the homogeneous coordinate ring of a smooth Calabi–Yau varieties X, the computation of Betti numbers can be reduced to computations of dimensions of certain \({\text {Hom}}\) spaces in the bounded derived category \(D^b(X)\). In the simplest case of a smooth elliptic curve E embedded in \({\mathbb {P}}^2\) as a smooth cubic, we get explicit values for Betti numbers. The 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 shifts in internal degree, and two possible shapes up to shifts in internal degree and taking syzygies.

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Acknowledgements

The results of the paper are part of the results of the author’s PhD thesis. I am pleased to thank my advisor Ragnar-Olaf Buchweitz, who posed the problem of computing Betti numbers in this case to me and encouraged me during my work on it. His interest in my results and numerous discussions on all stages of the project always motivated me.

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Correspondence to Alexander Pavlov.

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Pavlov, A. Betti tables of MCM modules over the cone of a plane cubic. Math. Z. 297, 223–254 (2021). https://doi.org/10.1007/s00209-020-02509-5

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Keywords

  • Betti numbers
  • Matrix Factorizations
  • Maximal Cohen–Macaulay modules
  • Plane Cubics
  • Elliptic curves

Mathematics Subject Classification

  • Primary 13D02
  • 13C14
  • 14H52
  • 14F05
  • Secondary 14H60
  • 14H50