Normal class and normal lines of algebraic hypersurfaces

Abstract

We are interested in the normal class of an algebraic hypersurface Z of the complex projective space P^n, that is the number of normal lines to Z passing through a generic point of P^n. Thanks to the notion of normal polar, we state a formula for the normal class valid for a general hypersurface Z of P^n. We give a generic result and we illustrate our formula with examples in P^n. We define the orthogonal indidence variety and compute the Schubert class of the variety of projective normal lines to a surface of P^3 in the Show ring of G(1,3). We complete our work with a generalization of Salmon's formula for the normal class of a Plucker curve to any planar curve with any kind of singularity.