  Let X be the base locus of a linear system W of k quadrics. Let also S be the
intersection of W with the discriminant hypersurface in the space of all
homogeneous polynomials of degree two. We prove a formula relating the topology
of X with the one of S and its (iterated) singular points. As a corollary we
prove the sharp bound b(X)\leq O(n)^{k-1}.
