Determinantal hypersurfaces

TL;DR

This paper explores when smooth hypersurfaces in projective space can be described by determinantal or pfaffian equations, linking this to the existence of specific vector bundles and providing computational results for low-dimensional cases.

Contribution

It establishes criteria for hypersurfaces to be defined by determinantal or pfaffian equations and demonstrates, via computational methods, the existence of such representations for general forms in low dimensions.

Findings

Hypersurfaces can be characterized by line bundles or vector bundles related to determinantal equations.

Computational evidence shows general forms in P^3 and P^4 can be expressed as pfaffians of skew-symmetric matrices.

Explicit bounds are provided for degrees where such representations exist.

Abstract

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 P^3 (resp. P^4) 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).