Generalizations of the odd degree theorem and applications

LetV ⊂ ℙℝn be an algebraic variety, such that its complexificationVℂ ⊂ ℙn is irreducible of codimensionm ≥ 1. We use a sufficient condition on a linear spaceL ⊂ ℙℝn of dimensionm + 2r to have a nonempty intersection withV, to show that any six dimensional subspace of 5 × 5 real symmetric matrices contains a nonzero matrix of rank at most 3.