Morphisms between Cremona groups, and characterization of rational varieties

Serge Cantat (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We classify all (abstract) homomorphisms from the group PGL(r+1)(C) to the group Bir(M) of birational transformations of a complex projective variety M, provided that r >= dim(C)(M). As a byproduct, we show that: (i) Bir(P-C(n)) is isomorphic, as an abstract group, to Bir(P-C(m)) if and only if n = m; and (ii) M is rational if and only if PGL(dim(M)+1)(C) embeds as a subgroup of Bir(M).

Mots clés: Cremona groups; rational varieties; algebraic transformation groups

Domaines: Géométrie algébrique [math.AG]; Théorie des groupes [math.GR]

Serge Cantat. Morphisms between Cremona groups, and characterization of rational varieties. Compositio Mathematica, 2014, 150 (7), pp.1107-1124. ⟨10.1112/S0010437X13007835⟩. ⟨hal-01057823⟩