Ideals of Spaces of Degenerate Matrices

Abstract

The variety Sing n,m consists of all tuples X = (X 1 , . . . , Xm) of n×n matrices such that every linear combination of X 1 , . . . , Xm is singular. Equivalently, X ∈ Sing n,m if and only if det(λ 1 X 1 +. . .+λmXm) = 0 for all λ 1 , . . . , λm ∈ Q. Makam and Wigderson [12] asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on Sing n,m lies in the ideal generated by the polynomials det(λ 1 X 1 + . . . + λmXm). We answer this question in the negative by determining the vanishing ideal of Sing 2,m for all m ∈ N. Our results exhibit that there are additional equations arising from the tensor structure of X. More generally, for any n and m ≥ n 2 − n + 1, we prove there are equations vanishing on Sing n,m that are not in the ideal generated by polynomials of type det(λ 1 X 1 + . . . + λmXm). Our methods are based on classical results about Fano schemes, representation theory and Gröbner bases.