Affine subspaces of matrices with rank in a range

For every m, n N and every field K, let M (m n, K) be the vector space of the (m n) -matrices over K and let A (n, K) be the vector space of the antisymmetric (n n) -matrices over K. Let r, s with s r. Define AK (m n;s, r) = \ array{ll S \;| S \; affine subspace of M (m n, K) such that A S \}, \; r = \rank (A) | \;, A S \ array \} A₀₍ₓ₈ₒₘ₌K (n;s, r) = \ array{ll S \;| S \; affine subspace of A (n, K) such that A S \}, \; r = \rank (A) | \; A S \ array \}, A₄₂₇₄₋K (m n;s, r) = \ array{ll S \;| S \; affine subspace of M (m n, K) such that \\ is in row echelon form A S A S \}, \; r = \rank (A) | \; A S \ array \}. In this paper we prove the following formulas: if |K| r+2 and the characteristic of K is different from 2, then \ (S) \; |\; S { AK (m n; s, r) \} = r \m, n\ - s+12 and \ (S) \; |\; S { A₀₍ₓ₈ₒₘ₌K (n; s, r) \} (n-1) r2 - s²4; if |K| m+1, then \ (S) \; |\; S { AK₄₂₇₄₋ (m n; r, r) \}= r n - r (r+1) 2.