Generalizations of the odd degree theorem and applications

Let V subset of PRn be in algebraic variety, such that is complexification V-C subset of P-n is irreducible of codimension m greater than or equal to 1. We use a sufficient, condition on a linear space L subset of PRn of dimension m + 2r to have a nonempty, intersection with V, to show that any six dimensional subspace of 5 x 5 real symmetric matrices contains a nonzero matrix of rank at most 3.