Topology of Blow-ups and Enumerative Geometry

Let M be the blow–up of a manifold M along a submanifold X. In this paper we present closed formulae for the integral cohomology and the total Chern class of M. As applications we compute the cohomology of the varieties of complete conics and complete quadrices in 3–space, and justify two enumerative results due to Schubert.