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Published online by Cambridge University Press: 29 May 2025
We describe the cone of Betti tables of Cohen–Macaulay modules over the homogeneous coordinate ring of a rational normal curve.
The study of the cone generated by the graded Betti tables of finitely generated modules over graded rings has received much attention recently. (See Definition 2.1 for the relevant definitions.) This began with a conjectural description of this cone in the case of polynomial rings by M. Boij and J. Söder-berg [2008] which was proved by D. Eisenbud and F.-O. Schreyer [2009]. We refer to [Eisenbud and Schreyer 2011; Fløystad 2012] for a survey of this development and related results. Similarly, in the local case, there is a description of the cone of Betti sequences over regular local rings [Berkesch et al. 2012b].
However, not much is known about the cone of Betti tables over other graded rings, or over nonregular local rings. The cone of Betti tables for rings of the form 𝕂[x, y]/q(x, y) where q is a homogeneous quadric is described in [Berkesch et al. 2012a]. In the local hypersurface case, [Berkesch et al. 2012b] gives some partial results and some asymptotic results. We also point to [Eisenbud and Erman 2012, Sections 9–10] for a study of Betti tables in the nonregular case.
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