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Let P X be a P¹-bundle over a variety X. The aim of this note is to understand all connected, algebraic groups Aut^ (P) G Bir (X P¹). We get a quite complete answer if Aut^ (X) is a maximal, connected, algebraic subgroup of Bir (X), and P is sufficiently unstable. This gives examples of connected, algebraic subgroups of Bir (P⁴) that are not contained in any maximal one.
Janós Kollár (Tue,) studied this question.