Let $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}_n(K)$, where $\textrm{Sym}_n(K)$ is the vector space of all symmetric $n \times n$ matrices with entries in $K$, such that every matrix in $M$ has at most rank $k$.

Are there any results on this problem? Is there a characterization of such subspaces?

If we do not restrict to symmetric matrices, but consider vector spaces of $m \times n$ matrices of rank at most $k$, this question was solved in the 80's for $k\leq 3$ by Atkinson and independently by Eisenbud and Harris. But the case $k>4$ is yet unknown, as far as I know. I think that if we restrict to symmetric matrices, this should be far less complicated, so perhaps there are results for $k>4$.

Thanks in advance!

On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual varieties, Math. Ann 314 (1999), 159-174. However, it doesn't quite answer your question, since they study subspaces in which nonzero elements have constant rank rather than bounded rank. However, the bibliography points to other papers that might be more directly relevant. $\endgroup$