Courbes rationnelles sur les vari\'et\'es homog\`enes et une d\'esingularisation plus fine des vari\'et\'es de Schubert

Abstract

In this article we prove the irreducibility of the Hilbert scheme of rationnal curves on homogeneous varieties with fixed class in the Chow ring. This result has also been proved by J. F. Thomsen [T] and B. Kim and R. Pandharipande [KP]. Our method is totaly different (we don't use the compactification of stable maps) and enables us to prove the existence of rational smooth curves on homogeneous varities with fixed class in the Chow ring. This was not the case of Thomsen's and Kim and Pandharipande's proofs. We use a decomposition of G/P in orbits (called the P'-orbits, see definition) which are bigger than the Schubert cells. We then prove that these P'-orbits are "towers" of affine bundles (see definition) over "smaller" homogeneous varities. This description gives the results. Our decomposition in P'-orbits enables us to give a "better" desingularisation of Schubert varities than Demazure's one.