1995/03/28 by Olivier Debarre, Debarre, Olivier · 2 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.alg-geom/9503020
Let G be the Grassmannian G(d,n), let X and Y be complete irreducible\nvarieties, and let X\→ G and Y\→ G be morphisms. Hansen\nproved that X \×G Y is connected when codim f(X) + codim g(Y) < n. We\nshow that the conclusion holds under the often weaker hypothesis\nf(X).g(Y).T\≠ 0, where T is the class of G(d,n-1) in G.\n We prove similar results when G is a product of projective spaces. In\nparticular, if D is an irreducible subvariety of Pn\× Pn of dimension\nn which dominates both factors, and if X is complete irreducible, with a\nmorphism f: X \→ Pn\× Pn such that dim f(X) >n, f-1(D)\nis connected. This extends the classical Fulton-Hansen connectedness theorem.\nThese results illustrate Fulton and Lazarsfeld's idea that connectedness should\nbe a numerical property.\n