Existence of Indecomposable Rank Two Vector Bundles on Higher Dimensional Toric Varieties

In the mid 1970s, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on ? ⁿ is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ? ⁿ .     

Riemann-Roch for Quotients and Todd Classes of Simplicial Toric Varieties

Abstract

In this paper we give an explicit formula for the Riemann-Roch map for singular schemes which are quotients of smooth schemes by diagonalizable groups. As an application we obtain a simple proof of a formula for the Todd class of a simplicial toric variety. An equivariant version of this formula was previously obtained for complete simplicial toric varieties by Brion and Vergne (Brion M. and Vergne M. ([1997] Brion, M. and Vergne, M. 1997. An equivariant Riemann-Roch theorem for complete simplicial toric varieties. J. Reine. Agnew. Math., 482: 67–92.  [Google Scholar]). An equivariant Riemann-Roch theorem for complete simplicial toric varieties. J. Reine. Agnew. Math.482:67–92) using different techniques.     

Combinatorial Tangent Space and Rational Smoothness of Schubert Varieties

Abstract

Following Contou-Carrère (Contou-Carrère,C. (1983). Géométrie des Groupes Semi-Simples,Résolutions équivariantes et Lieu Singulier de Leurs Variétés de Schubert. Thèse d’état,Université Montpellier II (published partly as,Le Lieu singulier des variétés de Schubert (1988). Adv. Math.,71:186–221)),we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the Tits building associated to the situation. Using Carrell and Peterson's characterization (Carrell,J. B. (1994). The Bruhat graph of a Coxeter group,a conjecture of Deodhar,and rational smoothness of Schubert varieties. Proc. Symp. in Pure Math. 56(Part I):53–61),we prove that rational smoothness of a Schubert variety can be expressed in terms of a subspace of the Zariski tangent space called,the combinatorial tangent space.     

Normal Sections and Direct Product Decompositions

Abstract

In any finitely complete category, there is an internal notion of normal monomorphism. We give elementary conditions guaranteeing that a normal section s: Y → X of an arrow f: X → Y produces a direct product decomposition of the form X ? Y × W. We then show how these conditions gradually vanish in various algebraic contexts, such as Maltsev, protomodular and additive categories.     

Dimension inequalities for the closure of the image of a multilinear function

In this note we give a simple proof and an extension of a dimension inequality of Howard concerning the range of a multilinear function with vector space range by using some results on algebraic varieties.     

Bundles over the Fano Threefold V 5

ABSTRACT

Using an explicit resolution of the diagonal for the variety V ₅, we provide cohomological characterizations of the universal and quotient bundles. A splitting criterion for bundles over V ₅ is also proved.

The presentation of semistable aCM bundles is shown, together with a resolution–theoretic classification of low rank aCM bundles.     

Complete Congruences on the Lattice of Rees–Sushkevich Varieties

We study the lattice ?(RS_n) of subvarieties of the variety of semigroups generated by completely 0-simple semigroups over groups with exponent dividing n, with a particular focus on the lattice ??(RS_n) consisting of those varieties that are generated by completely 0-simple semigroups. The sublattice of ??(RS_n) consisting of the aperiodic varieties is described and several endomorphisms of ?(RS_n) considered. The complete congruence on ??(RS_n) that relates varieties containing the same aperiodic completely 0-simple semigroups is considered in some detail.