Rational curves on minuscule Schubert varieties

Abstract

Introduction Let us denote by C the variety of lines in P3 meeting a fixed line, it is a grassmannian (and hence minuscule) Schubert variety. In [16] we described the irreducible components of the scheme of morphisms from P1 to C and the general morphism in each irreducible component. In this text we study the scheme of morphisms from P1 to any minuscule Schubert variety X. Let us recall that we studied in [15] the scheme of morphisms from P1 to any homogeneous variety. The main idea, in the case of a minuscule Schubert variety X, is to restrict ourselves to the dense orbit under the stabilizer Stab(X) of X and apply the results of [15]. More precisely, let U be the dense orbit under Stab(X) in X and let Y be the complement. Because X is a minuscule Schubert variety the closed subset Y of X is of codimension at least 2 (see Section 2.2). This fact and the stratification of X by Schubert subvarieties gives us a surjective morphism (see Section 1): s : Pic(U)∨ →A1(X).