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Paper here: 
http://arxiv.org/abs/2011.10871   

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 were $X$ has 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 codim(I) = 4 = regularity(I), and that 9 of these arise for prime nondegenerate threefolds. We investigate the situation in codimension five or more, obtaining examples of X with $h^{p,q}(X)$ not among those appearing for $I_X$ of lower codimension or as complete intersections in toric Fano varieties--in other words, Calabi-Yau's with Hodge numbers not previously known to occur. A main feature of our approach is the use of inverse systems to identify possible betti tables for X. This is joint work with  M. Stillman, B. Yuan.