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

TL;DR

This work develops a framework to realize affine $SL_2(k)$-varieties as equivariant quotients via minimal presentations, using locally nilpotent derivations and fundamental pairs to extend invariant rings without Reynolds operators. It provides an explicit, algorithmic method to obtain minimal presentations for all $SL_2(k)$-homogeneous threefolds, with detailed constructions for several finite subgroups via Klein’s invariants. The authors prove that the surfaces $Y=SL_2(k)/T$ and $X=SL_2(k)/N$ exhibit striking phenomena: $X$ has embedding dimension $4$ but a minimal presentation of dimension $5$, and its $SL_2(k)$-action is absolutely nonextendable (in particular, no closed embedding into $A_k^4$ exists); $X$ is noncancelative, and they settle the existence of inequivalent closed embeddings of $Y$ into $\mathbb{A}_k^3$. The cylinder over $X$ and the automorphism structure of $X$ and $Y$ are analyzed via amalgamated product descriptions, yielding rigidity results for $\mathbb{G}_a$-actions and shedding light on linearization-type questions in this setting.

Abstract

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.

Theorems & Definitions (32)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Example 1.5
• Example 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 2.1
• Lemma 2.2