Th\'eor\`emes De Connexit\'e Pour Les Produits D'Espaces Projectifs et Les Grassmanniennes

Let $G$ be the Grassmannian $G(d,n)$, let $X$ and $Y$ be complete irreducible varieties, and let $X\rightarrow G$ and $Y\rightarrow G$ be morphisms. Hansen proved that $X \times_G Y$ is connected when $codim f(X) + codim g(Y) < n$. 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 $f: X \rightarrow P^n\times P^n$ such that $dim f(X) >n$, $f^{-1}(D)$ 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.