53 research outputs found
Small resolutions of Schubert varieties and Kazhdan-Lusztig polynomials
This article does not have an abstract
Rational curves on minuscule Schubert varieties
Let X be a minuscule Schubert variety and a class of 1-cycle on X.
In this article we describe the irreducible components of the scheme of
morphisms of class from a rational curve to X.
The irreducible components are described in the following way : the class
can be seen as an element of the dual of the Picard group.
Because any Weil-divisor need not to be a Cartier-divisor, there is (only) a
surjective map 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 in such that
.
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 . 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 the stabiliser of X
and we can apply results of our previous paper math.AG/0003199.Comment: In english, 29 page
Richardson Varieties Have Kawamata Log Terminal Singularities
Let be a Richardson variety in the full flag variety associated
to a symmetrizable Kac-Moody group . Recall that is the intersection
of the finite dimensional Schubert variety with the finite codimensional
opposite Schubert variety . We give an explicit \bQ-divisor on
and prove that the pair has Kawamata log terminal
singularities. In fact, is ample, which additionally
proves that is log Fano.
We first give a proof of our result in the finite case (i.e., in the case
when is a finite dimensional semisimple group) by a careful analysis of an
explicit resolution of singularities of (similar to the BSDH resolution
of the Schubert varieties). In the general Kac-Moody case, in the absence of an
explicit resolution of as above, we give a proof that relies on the
Frobenius splitting methods. In particular, we use Mathieu's result asserting
that the Richardson varieties are Frobenius split, and combine it with a result
of N. Hara and K.-I. Watanabe relating Frobenius splittings with log canonical
singularities.Comment: 15 pages, improved exposition and explanation. To appear in the
International Mathematics Research Notice
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