On affine spaces of alternating matrices with constant rank

Let F be a field, and n≥r>0 be integers, with r even. Denote by An⁡(F) the space of all n-by-n alternating matrices with entries in F. We consider the problem of determining the greatest possible dimension for an affine subspace of An⁡(F) in which every matrix has rank equal to r (or rank at least r). Recently Rubei Affine subspaces of skewsymmetric matrices with constant rank. Linear Multilinear Algebra. 2023. in press. doi: 10.1080/03081087.2023.2198759. has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that n≥r+3 and |F|≥min(r−1,r2+2), we also determine the affine subspaces of rank r matrices in An⁡(F) that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case n≤r+2.