Geometrically rational real conic bundles and very transitive actions

In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.Comment: Compositio Mathematica (2010) To appea

Birational automorphisms of a three-dimensional double quadric with an elementary singularity

It is proved that the group of birational automorphisms of a three-dimensional double quadric with a singular point arising from a double point on the branch divisor is a semidirect product of the free group generated by birational involutions of a special form and the group of regular automorphisms. The proof is based on the method of `untwisting' maximal singularities of linear systems.Comment: 18 page

Halphen pencils on quartic threefolds

For every smooth quartic threefold, we classify all pencils on it whose general element is an irreducible surface birational to a smooth surface of Kodaira dimension zero.Comment: 20 page

On birational involutions of P3P^3

Let $X$ be a rationally connected three-dimensional algebraic variety and let $\tau$ be an element of order two in the group of its birational selfmaps. Suppose that there exists a non-uniruled divisorial component of the $\tau$-fixed point locus. Using the equivariant minimal model program we give a rough classification of such elements.Comment: 24 pages, late

Birational rigidity of a three-dimensional double cone

It is proved that a three-dimensional double cone is a birationally rigid variety. We also compute the group of birational automorphisms of such a variety. This work is based on the method of "untwisting" maximal singularities of linear system.Comment: 20 pages; AmsLaTe

Finite abelian subgroups of the Cremona group of the plane

This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We have thus to enumerate all the cases, which gives a long description, and then to show whether two cases are distinct or not, using some conjugacy invariants. For example, we use the non-rational curves fixed by one element of the group, and the action of the whole group on these curves. From this classification, we deduce a sequence of more general results on birational transformations, as for example the existence of infinitely many conjugacy classes of elements of order n, for any even number n, a result false in the odd case. We prove also that a root of some linear transformation of finite order is itself conjugate to a linear transformation.Comment: PHD Thesis, 189 pages, 34 figures, original text may be found at http://www.unige.ch/cyberdocuments/theses2006/BlancJ/meta.htm

Rationality problems and conjectures of Milnor and Bloch-Kato

We show how the techniques of Voevodsky's proof of the Milnor conjecture and the Voevodsky- Rost proof of its generalization the Bloch-Kato conjecture can be used to study counterexamples to the classical L\"uroth problem. By generalizing a method due to Peyre, we produce for any prime number l and any integer n >= 2, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree n unramified \'etale cohomology class with l-torsion coefficients. When l = 2, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified \'etale cohomology class of lower degree.Comment: 15 pages; Revised and extended version of http://arxiv.org/abs/1001.4574 v2; Comments welcome