Chandrasekhar Hall

The Singular tuples of matrices is not a null cone

K V Subrahmanyam

CMI

We say an m tuple of  $n \times n$ matrices $(X_1, X_2,\ldots,X_m)$ is a singular  tuple if  the (complex) linear  span  of $X_1,X_2,\ldots,X_m$
contains only matrices with determinant zero. A natural question is if there some reductive  group $G$ acting linearly on $C^{mn^2}$ such that
the  null  cone  for  the  action  is  precisely  singular  tuples  of matrices.  Recently Vishwambara  Makam and  Avi Wigderson  showed that
this is  not possible, if either  $m \geq 3$ or  $n \geq 3$. I  will give a sketch of their proof.