Kathlén Kohn: Coisotropic Hypersurfaces in Grassmannians

For every projective variety \(X \subseteq \mathbb{P}^n\) of dimension \(k\), the set of all projective subspaces of dimension \(n-k-1\) that intersect X is a hypersurface in the Grassmannian \(G(n-k-1, \mathbb{P}^n).\) This hypersurface is defined by a polynomial in the Plücker coordinates of \(G(n-k-1, \mathbb{P}^n),\) which is unique up to scaling and the Plücker relations. This polynomial is called the Chow form of X.

One can generalize this definition to projective subspaces of higher dimension: All subspaces of a fixed dimension that intersect X non-transversally form a subvariety of a Grassmannian, which is said to be coisotropic. We will study which coisotropic subvarieties are hypersurfaces, and we will show that the degrees of the coisotropic hypersurfaces are the well-studied polar degrees. Moreover, the coisotropic hypersurfaces of X and its projectively dual variety will be related, and we will see that all hyperdeterminants arise as coisotropic forms of Segre varieties. Finally, we will investigate how to recover the underlying projective variety X from a given coisotropic hypersurface and how to test if a given subvariety of a Grassmannian is coisotropic independently of X.