Degrés d’homogénéité de l’ensemble des intersections complètes singulières

Benoist, Olivier

Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 1189-1214.

Un résultat classique de Boole montre que, sur un corps de caractéristique 0, l’ensemble des hypersurfaces singulières de degré $d$ dans ${\mathbb{P}}^N$ est un diviseur de degré $(N+1){(d-1)}^N$ de l’espace projectif de toutes les hypersurfaces. On obtient ici des formules analogues pour des intersections complètes de codimension et de degrés quelconques dans ${\mathbb{P}}^N$, en toute caractéristique.

A classical result of Boole shows that, in characteristic $0$, the set of singular degree $d$ hypersurfaces in ${\mathbb{P}}^N$ 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 ${\mathbb{P}}^N$ of arbitrary codimension and degrees, in any characteristic.

Classification : 14M10, 14N05, 14M25
Mot clés : intersections complètes, dualité projective, variétés toriques, caractéristique finie
Keywords: Complete intersections, projective duality, toric varieties, finite characteristic

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