On the self-intersection cycle of surfaces and some classical formulas for their secant varieties

We study the relation between certain local and global numerical invariants of a projective surface given by the Stückrad-Vogel self-intersection cycle. In the smooth case we find a relation between the degrees of the tangent, the secant and the first polar variety which generalizes classical relations between the so called elementary invariants of surfaces. For singular surfaces this relation involves also the contributions of singularities to the self-intersection cycle. In order to make the formula really effective for singular surfaces, we need to describe the movable part of the self-intersection cycle on the singular locus. This is done explicitly for a particular class of non normal del Pezzo surfaces.