We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: ${\mathfrak{A}}_5$ , ${\text{PSL}}_2(\textbf{F}_7)$ , ${\mathfrak{A}}_6$ , ${\text{SL}}_2(\textbf{F}_8)$ , ${\mathfrak{A}}_7$ , ${\text{PSp}}_4(\textbf{F}_3)$ , ${\text{SL}}_2(\textbf{F}_{7})$ , $2.{\mathfrak{A}}_5$ , $2.{\mathfrak{A}}_6$ , $3.{\mathfrak{A}}_6$ or $6.{\mathfrak{A}}_6$ . All of these groups with a possible exception of $2.{\mathfrak{A}}_6$ and $6.{\mathfrak{A}}_6$ indeed act on some rationally connected threefolds.

We prove rationality criteria over nonclosed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$ . For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational varieties of this type.

Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over $\mathbb{Q}$ have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property. Unless explicitly stated otherwise, all varieties are assumed to be algebraic, geometrically irreducible and defined over an arbitrary field $\Bbbk$ of characteristic zero.

Let (X, C) be a germ of a 3-fold X with terminal singularities along an irreducible reduced complete curve C with a contraction f:(X, C) → (Z, o) such that C = f−1 (o)red and −KX is ample. Assume that (X, C) contains a point of type (IC) or (IIB). We complete the classification of such germs in terms of a general member containing C.

We classify del Pezzo surfaces with quotient singularities and Picard rank one which admit a ℚ-Gorenstein smoothing. These surfaces arise as singular fibres of del Pezzo fibrations in the 3-fold minimal model program and also in moduli problems.