Smooth embeddings of \(SL(2)\) and \(PGL(2)\) (Q1813947)

Let \(G\) be a connected algebraic group over a field \(k\) of characteristic 0, and \(H\) an algebraic subgroup. An ``embedding'' of the homogeneous space \(G/H\) is an algebraic normal variety with an action of \(G\) having an open dense orbit isomorphic to \(G/H\). Such an embedding is smooth if and only if the local rings of all orbits are regular. Luna and Vust have developed a method to classify embeddings of homogeneous spaces \(G/H\) where \(G\) is a connected reductive group. --- The author finds combinatorial conditions for embeddings of \(SL(2)\) and \(PGL(2)\) to be smooth. This paper makes a nice contribution to invariant theory.