In this paper, we prove the degenerations of Schubert varieties in a minusculeG/P, as well as the class of Kempf varieties in the flag varietySL(n)/B, to (normal) toric varieties. As a consequence, we obtain that determinantal varietes degenerate to (normal) toric varieties.

Degenerations of flag and Schubert varieties to toric varieties

Let Xr be a smooth Del Pezzo surface obtained from ℙ(2 by blowing up r ⩽8 points in general position. It is well known that for r e {3,4,5,6,7,8} the Picard group Pic(X r ) contains a canonical root system R r € A 2 × A1, A4, D5, E6, E7, E8. We prove some general properties of the Cox ring of X r (r ⩾ 4) and show its similarity to the homogeneous coordinate ring of the orbit of the highest weight vector in some irreducible representation of the algebraic group G associated with the root system R r .

The Cox Ring of a Del Pezzo Surface

De Concini, Kac and Procesi defined a family of subalgebras U + of a quantized universal enveloping algebra Uq(g), associated to the elements of the corresponding Weyl group W . They are deformations of the universal enveloping algebras U(n+ ∩ Adw(n−)) where n± are the nilradicals of a pair of dual Borel subalgebras. Based on results of Gorelik and Joseph and an interpretation of U + as quantized algebras of functions on Schubert cells, we construct explicitly the H invariant prime ideals of each U + and show that the corresponding poset is isomorphic to W, where H is the group of group-like elements of Uq(g). Moreover, for each H-prime of U w + we construct a generating set in terms of Demazure modules related to fundamental representations. Using results of Ramanathan and Kempf we prove similar theorems for vanishing ideals of closures of torus orbits of symplectic leaves of related Poisson structures on Schubert cells in flag varieties.

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Invariant prime ideals in quantizations of nilpotent Lie algebras

Let G be a connected semisimple algebraic group, B a Borel subgroup, T a maximal torus in B with Weyl group W, and Q a subgroup containing B. For 
$w \in W$
 , let 
$X_{wQ}$
 denote the Schubert variety 
$\overline{BwQ}/Q$
 . For 
$y\in W$
 such that 
$X_{yQ}\subseteq X_{wQ}$
 , one knows that ByQ / Q admits a T-stable transversal in 
$X_{wQ}$
 , which we denote by 
${\cal N}_{yQ,wQ}$
 . We prove that, under certain hypotheses, 
${\cal N}_{yQ, wQ}$
 is isomorphic to the orbit closure of a highest weight vector in a certain Weyl module. We also obtain a generalisation of this result under slightly weaker hypotheses. Further, we prove that our hypotheses are satisfied when Q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight, and 
$X_{yQ}$
 is an irreducible component of the boundary of 
$X_{wQ}$
 (that is, the complement of the open orbit of the stabiliser in G of 
$X_{wQ}$
 ). As a consequence, we describe the singularity of 
$X_{wQ}$
 along ByQ / Q and obtain that the boundary of 
$X_{wQ}$
 equals its singular locus.

Generic singularities of certain Schubert varieties

Our main result is the description of generators of the total coordinate ring of the blow-up of $P^n$ in any number of points that lie on a rational normal curve. As a corollary we show that the algebra of invariants of the action of a two-dimensional vector group introduced by Nagata is finitely generated by certain explicit determinants. We also prove the finite generation of the algebras of invariants of actions of vector groups related to T-shaped Dynkin diagrams introduced by Mukai.

Hilbert's 14th Problem and Cox Rings

Let ~q~ 1 . . . . , 2 ' , be invertible sheaves on a variety X. We may form the multigraded ring A = ~F(X, | where the vector (m) ranges in N". Then C = Spec(A) is a multi-cone. If X is a projective Cohen-Macaulay variety, n = l and L~'l is very ample, Serre has given a criterion for C to be Cohen-Macaulay in terms of the cohomology of sheaves on X. We know no reasonable analog of this result when n > 1. On the other hand we will give such a criterion for C to have rational singularity when X does. This is a stronger restriction on the singularities of C. Then we will give an application to multi-cones based on Schubert varieties. Let G be a reductive algebraic group, B a Borel subgroup and X ~G/B a Schubert variety. Let s176 &~ ..., 5q~ be the line bundles on G/B corresponding to the fundamental weights. An arbitrary line bundle corresponding to a dominant weight has the form

A. Ramanathan

Papers

Multi-cones over Schubert varieties