Actions of SL₂(k) on Affine k-Domains and Fundamental Pairs

Abstract

Working over a field k of characteristic zero, this paper studies algebraic actions of SL₂(k) on affine k-domains by defining and investigating fundamental pairs of derivations. There are three main results: (1) The Structure Theorem for Fundamental Derivations (Theorem 3.4) describes the kernel of a fundamental derivation, together with its degree modules and image ideals. (2) The Classification Theorem (Theorem 4.5) lists all normal affine SL₂(k)-surfaces with trivial units, generalizing the classification given by Gizatullin and Popov for complex $SL_{2}(\mathbb {C})$-surfaces. (3) The Extension Theorem (Theorem 7.6) describes the extension of a fundamental derivation of a k-domain B to B[t] by an invariant function. The Classification Theorem is used to describe three-dimensional UFDs which admit a certain kind of SL₂(k) action (Theorem 6.2). This description is used to show that any SL₂(k)-action on ${\mathbb {A}}_{k}^{3}$ is linearizable, which was proved by Kraft and Popov in the case k is algebraically closed. This description is also used, together with Panyushev’s theorem on linearization of SL₂(k)-actions on ${\mathbb {A}}_{k}^{4}$, to show a cancelation property for threefolds X: If k is algebraically closed, $X \times {\mathbb {A}}_{k}^{1} \cong {\mathbb {A}}_{k}^{4}$ and X admits a nontrivial action of SL₂(k), then $X \cong {\mathbb {A}}_{k}^{3}$ (Theorem 6.6). The Extension Theorem is used to investigate free $\mathbb {G}_{a}$-actions on ${\mathbb {A}}_{k}^{n}$ of the type first constructed by Winkelmann.