Nestings of Rational Homogeneous Varieties

Morphisms between Grassmannians, II

“…The notion of effective good divisibility of a smooth complex projective variety M (see [7,Section 2.1]) is related to the total codimension of effective zero divisors in the Chow ring A • (M ).…”

“…The effective good divisibility is known for the wide class of rational homogeneous manifolds: it was computed for Grassmannians in [8], for varieties of classical type independently in [5,7], for varieties of exceptional type in [5]. In this paper, we are mostly interested in Grassmannians G(k, n), parametrizing linear subspaces of dimension k in P n , for which e.d.…”

“…In [7], this idea was refined and reinterpreted geometrically, considering the Schubert varieties X H , X p ⊂ G(k, n), parametrizing linear spaces contained in a hyperplane H or passing through a point p; then [X H ] ∈ A k+1 (G(k, n)), [X p ] ∈ A n−k (G(k, n)), and clearly [X H ] • [X p ] = 0. If the pullback via ϕ of one of the two cycles is zero, one can construct a morphism from M to a smaller Grassmannian which factors via ϕ, allowing inductive arguments.…”