2,515 research outputs found
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
Small codimension subvarieties in homogeneous spaces
We prove Bertini type theorems for the inverse image, under a proper
morphism, of any Schubert variety in an homogeneous space. Using
generalisations of Deligne's trick, we deduce connectedness results for the
inverse image of the diagonal in where is any isotropic grassmannian.
We also deduce simple connectedness properties for subvarieties of . Finally
we prove transplanting theorems {\`a} la Barth-Larsen for the Picard group of
any isotropic grassmannian of lines and for the Neron-Severi group of some
adjoint and coadjoint homogeneous spaces.Comment: 20 page
Spherical multiple flags
For a reductive group G, the products of projective rational varieties
homogeneous under G that are spherical for G have been classified by
Stembridge. We consider the B-orbit closures in these spherical varieties and
prove that under some mild restrictions they are normal, Cohen-Macaulay and
have a rational resolution.Comment: 16 page
Study of some orthosymplectic Springer fibers
We decompose the fibers of the Springer resolution for the odd nilcone of the
Lie superalgebra \osp(2n+1,2n) into locally closed subsets. We use this
decomposition to prove that almost all fibers are connected. However, in
contrast with the classical Springer fibers, we prove that the fibers can be
disconnected and non equidimensional
Rational curves on homogeneous cones
Let G/Q be an homogeneous variety embedded in a projective space P thanks to
an ample line bundle L. Take a projective space containing P and form the cone
X over G/Q, we call this a cone over an homogeneous variety.
Let 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 situation depends on the line bundle L : if the projectivised tangent
space to the vertex contains lines (i.e. if G/Q contains lines in P) then the
irreducible components are described as in our paper math.AG/0407123 by the
difference between Cartier and Weil divisors. On the contrary if there is no
line in the projectivised tangent space to the vertex then there are new
irreducible components corresponding to the multiplicity of the curve through
the vertex.
As in math.AG/0407123 we use a resolution Y of X (the blowing-up) and study
the curves on Y.Comment: In english, 13 page
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