Affine subspaces of matrices with constant rank

For every $m,n in N$ and every field $K$, let $M(m imes n, K)$ be the vector space of the $(m imes n)$-matrices over $K$ and let $S(n,K)$ be the vector space of the symmetric $(n imes n)$-matrices over $K$. We say that an affine subspace $S$ of $M(m imes n, K)$ or of $S(n,K)$ has constant rank $r$ if every matrix of $S$ has rank $r$. Define $${cal A}^K(m imes n; r)= { S ;| ; S ; mbox{ m affine subsapce of $M(m imes n, K)$ of constant rank } r}$$ $${cal A}_{sym}^K(n;r)= { S ;| ; S ; mbox{ m affine subsapce of $S(n,K)$ of constant rank } r}$$ $$a^K(m imes n;r) = max {dim S mid S in {cal A}^K(m imes n; r ) }.$$ $$a_{sym}^K(n;r) = max {dim S mid S in {cal A}_{sym}^K(n,r) }.$$ In this paper we prove the following two formulas for $r leq m leq n$: $$a_{sym}^{R}(n,r) = r(n-r)+ left[rac{r^2}{4} ight],$$ $$a^{R}(m imes n,r) = r(n-r)+ rac{r(r-1)}{2} .$$