Ideals of Spaces of Degenerate Matrices

The variety $\mathrm{Sing}_{n, m}$ consists of all tuples $X = (X_1,\ldots, X_m)$ of $n\times n$ matrices such that every linear combination of $X_1,\ldots, X_m$ is singular. Equivalently, $X\in\mathrm{Sing}_{n,m}$ if and only if $\det(\lambda_1 X_1 + \ldots + \lambda_m X_m) = 0$ for all $\lambda_1,\ldots, \lambda_m\in \mathbb Q$. Makam and Wigderson asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on $\mathrm{Sing}_{n, m}$ lies in the ideal generated by the polynomials $\det(\lambda_1 X_1 + \ldots + \lambda_m X_m)$. We answer this question in the negative by determining the vanishing ideal of $\mathrm{Sing}_{2, m}$ for all $m\in \mathbb N$. Our results exhibit that there are additional equations arising from the tensor structure of $X$. More generally, for any $n$ and $m\ge n^2 - n + 1$, we prove there are equations vanishing on $\mathrm{Sing}_{n, m}$ that are not in the ideal generated by polynomials of type $\det(\lambda_1 X_1 + \ldots + \lambda_m X_m)$. Our methods are based on classical results about Fano schemes, representation theory and Gr\"obner bases.