Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 3

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

Pages: 764 – 792

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n3.a7

Authors

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.)

Abstract

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 $X$.

Published 22 February 2023