2020/11/21 by Hal Schenck, Schenck, Hal, Mike Stillman +3 · 1 citation
Mathematics · #14J32 (Primary) 13D02 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #math.AC #math.AG #msc:13D02 #msc:14J32
paper · pdf · doi:10.48550/arxiv.2011.10871
openalex publication_date 2020/11/21 · arxiv created 2021/08/10 · arxiv updated 2021/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A projectively normal Calabi-Yau threefold X ⊆ ℙn has an ideal IX which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when IX 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 IX is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when IX has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal I with codim(I)=4=reg(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 hp,q(X) not among those appearing for IX of lower codimension or as complete intersections in toric Fano varieties. A key tool in our approach is the use of inverse systems to identify possible betti tables for X.