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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!

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    $\begingroup$ One reference you might find helpful is Ilic, Landsberg, 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. $\endgroup$
    – user5117
    May 16 '14 at 13:46
  • $\begingroup$ Thank you for your references. Unfortunately, thats all slightly different than that what I am looking for... $\endgroup$
    – Hans
    May 23 '14 at 13:54
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There are three references you may want to check out:

  • R. Meshulam, On two extremal matrix problems, Linear Algebra Appl. 114/115 (1989), 261--271
  • R. Loewy, N. Radwan, Spaces of symmetric matrices of bounded rank, Linear Algebra Appl. 197-198 (1994), 189--215
  • R. Loewy, Large spaces of symmetric matrices of bounded rank are decomposable. Lin. Multlin. Alg. 48 (2001), 355--382.

In all those papers, the authors focus on subspaces of symmetric matrices with rank bounded above by $r$, and more precisely on the maximal dimension for these subspaces and on the structure of the spaces whose dimension is close to the maximal one.

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