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

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

1995-03-28

arxiv.org/abs/alg-geom/9503020v2

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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