Equations and syzygies of some Kalman varieties

TL;DR

This paper proves a conjecture about the minimal equations defining Kalman varieties for a specific subspace dimension, and describes their algebraic structure using exact sequences and resolutions.

Contribution

It confirms Ottaviani and Sturmfels' conjecture for dim L=3 and provides a detailed description of the minimal free resolution for dim L=2.

Findings

Proved the conjecture for dim L=3.

Described the minimal free resolution for dim L=2.

Proposed a general conjecture on exact sequences for all dim L.

Abstract

Given a subspace L of a vector space V, the Kalman variety consists of all matrices of V that have a nonzero eigenvector in L. Ottaviani and Sturmfels described minimal equations in the case that dim L = 2 and conjectured minimal equations for dim L = 3. We prove their conjecture and describe the minimal free resolution in the case that dim L = 2, as well as some related results. The main tool is an exact sequence which involves the coordinate rings of these Kalman varieties and the normalizations of some related varieties. We conjecture that this exact sequence exists for all values of dim L.