Calabi-Yau 3-folds in projective spaces and Gorenstein rings

Abstract

The defining ideal $I_X$ of a projectively normal Calabi-Yau 3-fold $X$ is arithmetically Gorenstein, of Castelnuovo-Mumford regularity 4. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ has codimension 3. 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 4. We prove there are 16 possible Betti tables for an arithmetically Gorenstein ideal with codimension 4 and regularity 4, and that 8 of these arise for prime nondegenerate 3-folds. A main feature of our approach is the use of inverse systems to identify possible Betti tables for $X$. This is a joint work with H. Schenck and M. Stillman.