Rational curves on minuscule Schubert varieties

Abstract

Let X be a minuscule Schubert variety and $\alpha$ a class of 1-cycle on X. In this article we describe the irreducible components of the scheme of morphisms of class $\alpha$ from a rational curve to X. The irreducible components are described in the following way : the class $\alpha$ 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 $s:A^1(X)^*\to Pic(X)^*$ 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 $\beta$ in $A^1(X)^*$ such that $s(\beta)=\alpha$. 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 $Y\to X$. 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.