Nonlinear Kalman Varieties

Given a projective variety $X\in\mathbb{P}^{n-1}$, the associated Kalman variety $K(X)$ is defined as the set of projective matrices with at least an eigenvector in $X$.

Classically, the attention has been focused on linear varieties. In this talk, we explore the nonlinear case by studying the basic invariants of these varieties, such as their dimensions, degrees, and singularities.

A central result in the linear case is the existence of determinantal equations. We generalize this by providing a "determinantal-like" description for the equation of Kalman varieties associated with hypersurfaces.

This talk is based on a joint work with Luca Sodomaco and Julian Weigert.