13 research outputs found
On -birational rigidity of del Pezzo surfaces
No full textLet be a finite group and be its subgroup. We prove that
if a smooth del Pezzo surface over an algebraically closed field is
-birationally rigid then it is also -birationally rigid, answering a
geometric version of Koll\'{a}r's question in dimension 2 by positive.Comment: 27 pages. Substantially updated version, main results unchanged. A
new section is added, plus many explicit example
The Jordan constant for Cremona group of rank 2
No full textWe compute the Jordan constant for the group of birational automorphisms of a
projective plane , where is either an
algebraically closed field of characteristic 0, or the field of real numbers,
or the field of rational numbers.Comment: 12 page
On -birational rigidity of del Pezzo surfaces
No full textLet be a finite group and be its subgroup. We prove that if a smooth del Pezzo surface over an algebraically closed field is -birationally rigid then it is also -birationally rigid, answering a geometric version of Koll\'{a}r's question in dimension 2 by positive
Quotients of groups of birational transformations of cubic del Pezzo fibrations
No full textWe prove that the group of birational transformations of a del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2 . As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, the subgroup of Bir ( P 3 ) generated by all connected algebraic subgroups is a proper normal subgroup
Iterated polynomials are dense
No full textInternational audienceFor any infinite field k and any positive integer r, we show constructively that the map sending each polynomial P ∈ k[x] to its r-th iterate is dominant in various inductive limit topologies on the space of all polynomials.Pour tout corps infini k et tout entier r>0, nous montrons de manière constructive que l'application qui à P ∈ k[x] associe son itéré r-ième est d'image dense, et ce pour diverses topologies sur k[x] obtenues par limite inductive
Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank
No full textWe prove that any group of cardinality at most the one of is a quotient of any Cremona group of rank at least . This provides a definitive answer to the question of what the quotients of Cremona groups can be. As a consequence, this gives a negative answer to the question of I. Dolgachev of whether Cremona groups of all ranks are generated by involutions. As another application, we show that higher Cremona groups do not enjoy some classical group-theoretic properties (namely, the Hopfian property) which are satisfied by Cremona groups of rank . Finally, we discover that the -torsion of the Cremona group of rank at least is not countable.
To deduce these properties of higher Cremona groups, we first describe the group of birational transformations of a non-trivial Severi-Brauer surface over a perfect field, proving in particular that if contains a point of degree , then its group of birational self-maps is not generated by elements of finite order as it admits a surjective group homomorphism to~. We then use this result to study Mori fibre spaces over the field of complex numbers, for which the generic fibre is a non-trivial Severi-Brauer surface.72 page
Linearization problem for finite subgroups of the plane Cremona group
No full textWe give a complete solution of the linearization problem in the plane Cremona group over an algebraically closed field of characteristic zero
