Generalized Plücker Formulas

The classical Plücker formula for a plane curve was generalized by Teissier to the case of a hypersurface with isolated singularities and further by Kleiman to the case of an arbitrary n-dimensional projective variety V with isolated singularities. The formula relates the zeroth rank of V (the degree of the dual variety) to the Segre numbers of the conormal module and certain Buchsbaum—Rim multiplicities associated to the singular points of V. A second generalization was obtained by Pohl. It relates the (n-1)th rank of V to the first Chern class of a desingularization of V and the degree of the cuspidal divisor. We describe, for a projective variety V with arbitrary singularities, a natural class in the Chow group of the singular locus whose top dimensional part is given by Buchsbaum—Rim multiplicities, and we obtain generalizations of both formulas. The formulas are equations in the Chow group of V. They imply numerical formulas for all the ranks of V.