Presentations, embeddings and automorphisms of homogeneous spaces for SL(2,C)

For an algebraically closed field $k$ of characteristic zero and a linear algebraic $k$-group $G$, it is well known that every affine $G$-variety admits a $G$-equivariant closed embedding into a finite-dimensional $G$-module. Such an embedding is a presentation of the $G$-variety, and a minimal presentation is one for which the dimension the $G$-module is minimal. The problem of finding a minimal presentation generalizes the problem of determining whether a group action on affine space is linearizable. We give a minimal presentation for each homogeneous space for $SL_2(k)$. This constitutes the paper's main work. Of particular interest are the surfaces $Y=SL_2(k)/T$ and $X=SL_2(k)/N$ where $T$ is the one-dimensional torus and $N$ is its normalizer. We show that the minimal presentation of $X$ has dimension 5, the embedding dimension of $X$ is 4, and there does not exist a closed $SL_2$-equivariant embedding of $X$ in $A_k^4$. Thus, the $SL_2$-action on $X$ is absolutely nonextendable to $A_k^4$. We give two other examples of surfaces with absolutely nonextendable group actions. In addition, $X$ is noncancelative, that is, there exists a surface $Z$ such that $X\times A_k^1\cong_k Z\times A_k^1$ and $X\not\cong_kZ$. Finally, we settle the long-standing open question of whether there exist inequivalent closed embeddings of $Y$ in $A_k^3$ by constructing inequivalent embeddings.