Pascal Fong

The classification of algebraic subgroups of groups of
birational transformations have been initiated by the Italian school
of algebraic geometry. More than a hundred years ago, Enriques and
Fano have listed the maximal connected algebraic subgroups of the
group of birational transformations of $\mathbb{P}^3$. A proof of
their classification has been given by Umemura, through a series of
four articles and using analytic methods. More recently, Blanc,
Fanelli and Terpereau have mostly recovered this classification using
an algebraic approach. Let $C$ be a smooth projective curve of
positive genus. In this talk, we follow the strategy of Blanc, Fanelli
and Terpereau to study the pairs $(X,\mathrm{Aut}^\circ(X))$, where
$X$ is a $\mathbb{P}^1$-bundle over a ruled surface $S$ (with $S$
birational to $C\times \mathbb{P}^1$) and $\mathrm{Aut}^\circ(X)$ is a
maximal connected algebraic subgroup of $\mathrm{Bir}(C\times
\mathbb{P}^2)$.