Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 3

Calabi–Yau threefolds in $\mathbb{P}^n$ and Gorenstein rings

Pages: 764 – 792



Hal Schenck (Mathematics Department, Auburn University, Auburn, Alabama, U.S.A.)

Mike Stillman (Mathematics Department, Cornell University, Ithaca, New York, U.S.A.)

Beihui Yuan (Mathematics Department, Cornell University, Ithaca, New York, U.S.A.)


A projectively normal Calabi–Yau threefold $X \subseteq \mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo–Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum–Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are $16$ possible betti tables for an arithmetically Gorenstein ideal $I$ with $\operatorname{codim}(I) = 4 = \operatorname{regularity}(I)$, and that exactly $8$ of these occur for smooth irreducible nondegenerate threefolds. We investigate the situation in codimension five or more, obtaining examples of $X$ with $^{p,q} (X)$ not among those appearing for $I_X$ of lower codimension or as complete intersections in toric Fano varieties. A key tool in our work is the use of inverse systems to identify possible betti tables for $X$.

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Schenck supported by NSF 1818646. Stillman and Yuan supported by NSF 1502294.

Published 22 February 2023