Abstract

We give a numerical condition on the images of two morphisms to a Grassmannian (or a product of projective spaces) that ensures that their fibered product is connected, thereby extending connectedness results of Fulton and Hansen. This result is valid over any algebraically closed field; it yields a condition on the class of an irreducible subvariety of a Grassmannian that implies that it is simply connected. This applies in particular to Fano varieties of certain hypersurfaces in a projective space.

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