Large spaces of bounded rank matrices revisited

Let $n,p,r$ be positive integers with $n \geq p\geq r$. A rank-$\overline{r}$ subset of $n$ by $p$ matrices (with entries in a field) is a subset in which every matrix has rank less than or equal to $r$. A classical theorem of Flanders states that the dimension of a rank-$\overline{r}$ linear subspace must be less than or equal to $nr$, and it characterizes the spaces with the critical dimension $nr$. Linear subspaces with dimension close to the critical one were later studied by Atkinson, Lloyd and Beasley over fields with large cardinality; their results were recently extended to all fields. Using a new method, we obtain a classification of rank-$\overline{r}$ affine subspaces with large dimension, over all fields. This classification is then used to double the range of (large) dimensions for which the structure of rank $\overline{r}$-linear subspaces is known for all fields.