Actions of $SL_2(k)$ on affine $k$-domains and fundamental pairs

Abstract

Working over a field $k$ of characteristic zero, this paper studies algebraic actions of $SL_2(k)$ 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 $SL_2(k)$-surfaces with trivial units, generalizing the classification given by Gizatullin and Popov for complex $SL_2(C)$-surfaces [16]. (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 $SL_2(k)$-action (Theorem 6.2). This description is used to show that any $SL_2(k)$-action on $A_k^3$ 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 $SL_2(k)$-actions on $A_k^4$, to show a cancelation property for threefolds $X$: If $k$ is algebraically closed, $X\times A_k^1\cong A_k^4$ and $X$ admits a notrivial action of $SL_2(k)$, then $X\cong A_k^3$ (Theorem 6.6). The Extension Theorem is used to investigate free $G_a$-actions on $A_k^n$ of the type first constructed by Winkelmann.