Degr\'es d'homog\'en\'eit\'e de l'ensemble des intersections compl\`etes singuli\`eres

TL;DR

This paper generalizes classical formulas for the degree of the set of singular hypersurfaces to complete intersections of arbitrary codimension and degrees in projective space, applicable in any characteristic.

Contribution

It provides new formulas for the degrees of singular complete intersections in projective space, extending Boole's classical results beyond hypersurfaces and characteristic zero.

Findings

Formulas for degrees of singular complete intersections in P^N

Applicable in any characteristic, not just characteristic zero

Extension of Boole's classical hypersurface results

Abstract

A classical result of Boole shows that, in characteristic 0, the set of singular degree d hypersurfaces in P^N is a divisor of degree (N+1)(d-1)^N in the projective space of all hypersurfaces. We give here analogous formulae for complete intersections in P^N of arbitrary codimension and degrees, in any characteristic. ----- Un resultat classique de Boole montre que, sur un corps de caracteristique 0, l'ensemble des hypersurfaces singulieres de degre d dans P^N est un diviseur de degre (N+1)(d-1)^N de l'espace projectif de toutes les hypersurfaces. On obtient ici des formules analogues pour des intersections completes de codimension et de degres quelconques dans P^N, en toute caracteristique.