and

Let V ⊂ PRn be an algebraic variety, such that its complexifi-cation VC ⊂ Pn is irreducible of codimension m ≥ 1. We use a sufficient condition on a linear space L ⊂ PRn of dimension m + 2r to have a nonempty intersection with V, to show that any six dimensional subspace of 5 × 5 real symmetric matrices contains a nonzero matrix of rank at most 3.