Equations and syzygies of some Kalman varieties

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.Introduction.

Tensors with eigenvectors in a given subspace

“…Remark 3.12. In the matrix case d = 2, equations for κ s d,n,m (L) were found in [20,10], even in the more general case of matrix eigenvectors. It should be interesting to extend that study to the tensor case.…”

Matrices with eigenvectors in a given subspace

“…Theorem 3.6. (Sam [15]) The ideal I 3,n is minimally generated by polynomials of degree 3, 4, 5, 6. There are n−3 3 generators in degree 3, there are 2 n−2 3 generators each in degree 4 and in degree 5, and there are n−1 3 generators in degree 6.…”

“…Remark 5.3. After this paper had been written and refereed, we received Steven Sam's preprint [15] which uses Theorem 5.2 and the Weyman method to obtain the ideal sheaves of K 2,n and K 3,n , among other many results. In particular, Sam proved our conjecture (Theorem 3.6) on the defining equations of K 3,n .…”