Smooth embeddings of SL(2) and PGL(2)

Abstract

An embedding of an algebraic homogeneous space G/H is an algebraic normal variety with an action of G having an open (dense) orbit isomorphic to G/H. There is a beautiful theory for the embeddings of tori. See, for example, [KKMS], [Danl], or [Oda]. In this theory the embeddings are classified by “fans” of rational convex cones. By studying the combinatorics of the fans, one can determine certain geometric properties of the associated embeddings. For example, one can check which embeddings are smooth, complete, or projective. Recently Luna and Vust developed a method to classify embeddings of homogeneous spaces G/H where G is a connected algebraic reductive group [LV]. In the present paper we analyse more closely the cases of embeddings of X(2) and PGL(2). Each embedding is represented by a diagram containing combinatorical information about the local rings of orbits. As in the case of torus embeddings, one would like to be able to determine geometric properties from these diagrams. In this article we find the conditions that the diagrams must satisfy to represent smooth embeddings. The idea of the proof is as follows. An embedding is smooth if and only if the local rings of all the orbits are regular. The embedding is a three-dimensional normal variety, so all orbits of dimension two are smooth. It can be shown that the fixed points are never smooth, so one is left to treat the one-dimensional orbits (which are isomorphic to the projective line). A result from [BLV] states that a point of a one-dimensional orbit has an affme neighborhood which is the product of the affrne line and a certain two-dimensional variety; the orbit projects down to a point in the two-dimensional variety. Thus the problem is reduced to checking the regularity of certain local rings of points in twodimensional varieties. In each case, either one can find explicitly two