Coisotropic Hypersurfaces in Grassmannians

TL;DR

This paper introduces coisotropic hypersurfaces associated with projective varieties, providing new insights into their properties, dualities, degrees, and connections to hyperdeterminants, along with computational tools.

Contribution

It offers a new proof of the characterization of coisotropic hypersurfaces, explores their duality and degree properties, and links them to hyperdeterminants and the Cayley variety, with computational implementation.

Findings

Coisotropic hypersurfaces of a variety and its dual are equal.

Degrees of coisotropic hypersurfaces are the polar degrees of the variety.

Hyperdeterminants define coisotropic hypersurfaces of Segre varieties.

Abstract

To every projective variety $X$, we associate a family of hypersurfaces in different Grassmannians, called the coisotropic hypersurfaces of $X$. These include the Chow form and the Hurwitz form of $X$. Gel'fand, Kapranov and Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on tangent spaces. We present a new and simplified proof of that result. We show that the coisotropic hypersurfaces of $X$ equal those of its projectively dual variety, and that their degrees are the polar degrees of $X$. Coisotropic hypersurfaces of Segre varieties are defined by hyperdeterminants, and all hyperdeterminants arise in that manner. We derive new equations for the Cayley variety which parametrizes all coisotropic hypersurfaces of given degree in a fixed Grassmannian. We provide a Macaulay2 package for transitioning between $X$ and its coisotropic hypersurfaces.