Théorèmes De Connexité Pour Les Produits D'Espaces Projectifs et Les Grassmanniennes

Abstract

Let $G$ be the Grassmannian $G(d,n)$, let $X$ and $Y$ be complete irreducible varieties, and let $X\to G$ and $Y\to G$ be morphisms. Hansen proved that $X{\times }_{G}Y$ is connected when $codimf(X)+codimg(Y)<n$. We show that the conclusion holds under the often weaker hypothesis $f(X).g(Y).T\ne 0$, where $T$ is the class of $G(d,n-1)$ in $G$. We prove similar results when $G$ is a product of projective spaces. In particular, if $D$ is an irreducible subvariety of $P^n\times P^n$ of dimension $n$ which dominates both factors, and if $X$ is complete irreducible, with a morphism $f:X\to P^n\times P^n$ such that $dimf(X)>n$, $f^{-1}(D)$ is connected. This extends the classical Fulton-Hansen connectedness theorem. These results illustrate Fulton and Lazarsfeld's idea that connectedness should be a numerical property.

In French, 24 pages. The file originally submitted was corrupted. This one should work fine. PlainTeX v 1.2 with amssym.def and amssym

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