Estimates on the Distribution of the Condition Number of Singular Matrices

We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bézout Equality for equi–dimensional varieties. We also state an upper bound for the volume of a tube about a projective variety. As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with an equi–dimensional projective algebraic variety. We apply these techniques to exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices.