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

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