Rocco Chirivì

LS algebras and application to Schubert varieties

In this paper we introduce LS algebras. We study their general properties and apply these results to Schubert varieties. Our main achievement is that any Schubert variety admits a flat deformation to a union of normal toric varieties. A new proof of Cohen-Macaulayness (and thus normality) for Schubert varieties is also obtained.

Projective normality of complete symmetric varieties

We prove that in characteristic zero the multiplication of sections of line bundles generated by global sections on a complete symmetric variety X = G/H is a surjective map. As a consequence, the cone defined by a complete linear system over X or over a closed G-stable subvariety of X is normal. This gives an affirmative answer to a question raised by Faltings in [11]. A crucial point of the proof is a combinatorial property of root systems. Introduction Let G be an adjoint semisimple algebraic group over an algebraically closed field of characteristic zero, and let G be its algebraic simply connected cover. Given an involutorial automorphism σ : G → G, denote by H the subgroup of fixed points of σ . A wonderful compactification X of the symmetric variety G/H has been constructed by De Concini and Procesi [9]. The main result of our paper can be stated as the following. THEOREM A If L and L ′ are line bundles generated by global sections on X , then the multiplication 0(X,L )⊗ 0(X,L )→ 0(X,L ⊗L ) is surjective. The projective normality of X follows by a standard argument. Hence we give an affirmative answer to a problem raised by Faltings in [11]. Our result has already been proved in [15] by Kannan in the special case of the compactification of a group, in which G = H×H and the involution exchanges the two copies of H , by a completely different method that does not apply to this situation. We stress that it is necessary to assume that the line bundles L and L ′ are generated by global sections, as the example after the proof of Theorem A in Section 3 shows. DUKE MATHEMATICAL JOURNAL Vol. 122, No. 1, c

The ring of sections of a complete symmetric variety

We study the ring of sections A(X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semisimple group and H is the fixed subgroup for an involutorial automorphism of G. We find generators for Pic (X), we generalize the PRV conjecture to complete symmetric varieties and construct a standard monomial theory for A(X) that is compatible with G orbit closures in X. This gives a degeneration result and the rational singularityness for A(X).  2003 Elsevier Science (USA). All rights reserved.

Deformation and Cohen-Macaulayness of the multicone over the flag variety

Abstract. A general theory of LS algebras over a multiposet is developed. As a main result, the existence of a flat deformation to discrete algebras is obtained. This is applied to the multicone over partial flag varieties for Kac-Moody groups proving a deformation theorem to a union of toric varieties. In order to achieve the Cohen-Macaulayness of the multicone we show that Bruhat posets (defined as glueing of minimal representatives modulo parabolic subgroups of a Weyl group) are lexicographically shellable.

Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical non-Hermitian cases

Given a classical semisimple complex algebraic group G and a symmetric pair (G, K) of non-Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. For all such orbit closures, we study the normality and we describe the K-module structure of the ring of regular functions of the normalizations.

Standard Monomial Theory for Wonderful Varieties

A general setting for a standard monomial theory on a multiset is introduced and applied to the Cox ring of a wonderful variety. This gives a degeneration result of the Cox ring to a multicone over a partial flag variety. Further, we deduce that the Cox ring has rational singularities.

A Note on Normality of Cones Over Symmetric Varieties

Let G be a semisimple and simply connected algebraic group, and let H 0 be the subgroup of points fixed by an involution of G. Let V be an irreducible representation of G with a nonzero vector v fixed by H 0. In this article, we prove a property of the normalization of the coordinate ring of the closure of G·[v] in ℙ(V).

On exceptional completions of symmetric varieties

We search for the charmed pentaquark candidate reported by the H1 collaboration, the Θ_c(3100)^0, in e^+e^- interactions at a center-of-mass (c.m.) energy of 10.58 GeV, using 124  fb^(-1) of data recorded with the BABAR detector at the PEP-II e^+e^- facility at SLAC. We find no evidence for such a state in the same pD^(*-) decay mode reported by H1, and we set limits on its production cross section times branching fraction into pD^(*-) as a function of c.m. momentum. The corresponding limit on its total rate per e^+e^-→qq event, times branching fraction, is about 3 orders of magnitude lower than rates measured for the charmed Λ_c and Σ_c baryons in such events.

A relation on minimal representatives and the path model

We study a relation defined in terms of the Bruhat order on minimal representatives modulo parabolic subgroups of a finite Coxeter group. This relation generalizes the set inclusion for standard rows on type Al. As an application, we give a combinatorial description of Littelmanns swap map for minuscule path models.