Rational curves on minuscule Schubert varieties

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Let X be a minuscule Schubert variety and α a class of 1-cycle on X. In this article we describe the irreducible components of the scheme of morphisms of class α from a rational curve to X. The irreducible components are described in the following way: the class α can be seen as an element of Pic(X)* the dual of the Picard group. Because any Weil-divisor need not to be a Cartier-divisor, there is (only) a surjective map s:A¹(X)*→ Pic(X)* from the dual of the group of codimension 1 cycles to the dual of the Picard group. The irreducible components are given by the effective elements β in A¹(X)* such that s(β)=α. The proof of the result uses the Bott-Samelson resolution Y of X. We prove that any curve on X can be lifted in Y (after deformation). This is because any divisor on minuscule Schubert variety is a moving one. Then we prove that any curve coming from X can be deformed so that it does not meet the contracted divisor of Y→ X. This is possible because for minuscule Schubert variety there are lines in the projectivised tangent space to a singularity. It is now sufficient to deal with the case of the orbit of Stab(X) the stabiliser of X and we can apply results of our previous paper math.AG/0003199.