, Volume 1, Issue 4, pp 375-403

Wonderful varieties of rank two

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Abstract

LetG be a complex connected reductive group. Well known wonderfulG-varieties are those of rank zero, namely the generalized flag varietiesG/P, those of rank one, classified in [A], and certain complete symmetric varieties described in [DP] such as the famous space of complete conics. Recently, there is a renewed interest in wonderful varieties of rank two since they were shown to hold a keystone position in the theory of spherical varieties, see [L], [BP], and [K].

The purpose of this paper is to give a classification of wonderful varieties of rank two. These are nonsingular completeG-varieties containing four orbits, a dense orbit and two orbits of codimension one whose closuresD 1 andD 2 intersect transversally in the fourth orbit which is of codimension two. We have gathered our results in tables, including isotropy groups, explicit basis of Picard groups, and several combinatorial data in relation with the theory of spherical varieties.