1999/10/06 by Arnaud Beauville, A. Beauville, Beauville, A. · 9 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG
paper · pdf · doi:10.48550/arxiv.math/9910030
29 pages, Plain TeX, with a new Appendix by F.-O. Schreyer
openalex publication_date 1999/10/06 · arxiv created 2000/05/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a smooth hypersurface in projective space. We discuss in this paper when X can be defined by an equation det M = 0 (resp. pf M = 0), where M is a matrix (resp. a skew-symmetric matrix) with homogeneous entries. Standard homological algebra methods show that this is equivalent to produce a line bundle (resp. a rank 2 vector bundle) E of a certain type on X . We discuss a number of applications for hypersurfaces of small dimension. An Appendix by F.-O. Schreyer proves (using Macaulay 2) that a general form of degree d in P3 (resp. P4) can be written as the pfaffian of a skew-symmetric (2d)x(2d) matrix with linear entries in the expected range, that is d < 16 (resp. d < 6).