Calabi-Yau threefolds in P^n and Gorenstein rings

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 and B. Yuan

commutative algebra

Audience: researchers in the topic

Fellowship of the Ring

Series comments: Description: National Commutative Algebra Seminar

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