Maximal subgroups in the Cremona group

Abstract

We show that for any $n\ge 5$ there exist connected algebraic subgroups in the Cremona group $\mathrm{Bir}({\mathbb{P}}^n)$ that are not contained in any maximal connected algebraic subgroup. Our approach exploits the existence of stably rational, non-rational threefolds.

Mistake in proof of Proposition 2.1 is fixed, added Appendix