Abstract:
In 1893, Enriques showed that in the group $\mathrm{Bir}(\mathbb{P}^2)$ every
connected algebraic subgroup is conjugate to a subgroup of $\mathrm{Aut}^
\circ(\mathbb{P}^2)$ or of $\mathrm{Aut}^\circ(\mathbb{F}^n)$ for $n\geq 2$.
These automorphism groups are maximal in terms of inclusion. Umemura
classified connected subgroups in $\mathbb{P}^3$ and it also turned out that
every connected subgroup is contained in the maximal one. Blanc formulated the
following question: is it true that such a property is true for Cremona groups
in arbitrary dimension? Fong showed that in the group $\mathrm{Bir}(C\times
\mathbb{P}^1)$ there are connected algebraic subgroups that are not contained
in a maximal one. Fanelli, Floris and Zimmermann answered negatively to
Blanc's question in dimension $\geq 5$. I will tell about these results,
following the article of the latter authors and Fong's articles.