Classification of two-orbit varieties

We obtain the classification of two-orbit varieties, i.e. the normal complete complex algebraic varieties on which a reductive complex algebraic group acts with two orbits. We prove also Luna's conjecture saying that these varieties are spherical, i.e. admit a dense orbit of a Borel subgroup. Received: September 1, 2000     

Principal bundles on the affine line

We prove that any principal bundle on the affine line over a perfect field with a reductive group as structure group comes from the base field by base change.     

The Characteristics of Direct Hydroxylation of Benzene to Phenol with Molecular Oxygen Enhanced by Pulse DC Corona at Atmospheric Pressure

The direct hydroxylation of benzene using molecular oxygen by atmospheric pulse DC corona discharge was investigated. The conversion of benzene increased with the increase of oxygen content and input voltage but the selectivity of phenol decreased due to the formation of polymerized products. The reactivity was also influenced by the kind and content of background inert gas. By using argon as background gas, we could get 2.2% of phenol yield at 60°C and 1 atm with energy consumption of 50 W. The strategy of reductive oxidation, which added hydrogen to the reactant, was not favorable to the phenol formation in this reaction system. The polymerized product showed the oligomeric character and the analysis of its chemical structure with FT–IR was presented.     

Maximal Parabolic Subgroups in Classical Groups are of Modality Zero

The principal aim of this paper is to show that every maximal parabolic subgroup P of a classical reductive algebraic group G operates with a finite number of orbits on its unipotent radical. This is a consequence of the fact that each parabolic subgroup of a group of type A _n whose unipotent radical is of nilpotent class at most two has this finiteness property.     

A variant of the Reynolds operator

Let be a linearly reductive group over a field , and let be a -algebra with a rational action of . Given rational --modules and , we define for the induced -action on Hom a generalized Reynolds operator, which exists even if the action on Hom is not rational. Given an -module homomorphism , it produces, in a natural way, an -module homomorphism which is -equivariant. We use this generalized Reynolds operator to study properties of rational - modules. In particular, we prove that if is invariantly generated (i.e. ), then is a projective (resp. flat) -module provided that is a projective (resp. flat) -module. We also give a criterion whether an -projective (or -flat) rational --module is extended from an -module.

     

Spherical Tropical Geometry: a Survey of Recent Developments

This is a survey of some recent results on spherical tropical geometry.