On the normal class of curves and surfaces

Alfrederic Josse (1) , Françoise Pene (1)

1 LM - Laboratoire de mathématiques de Brest

We are interested in the normal class of an algebraic surface S of the complex projective space P^3, that is the number of normal lines to S passing through a generic point of P^3. Thanks to the notion of normal polar, we state a formula for the normal class valid for a general surface S. We give a generic result and we illustrate our formula with examples. 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. This last formula gives directly the normal class of any cylinder and of any surface of revolution of P^3.

Mots clés: projective normal; class; Plücker; Grassmannian; Puiseux

Domaines: Géométrie algébrique [math.AG]

Alfrederic Josse, Françoise Pene. On the normal class of curves and surfaces. 2014. ⟨hal-00953669v1⟩