Related papers: Birational automorphisms of nodal quartic threefol…

We construct families of birational involutions on $\mathbb{P}^3$ or a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of…

We prove birational rigidity and calculate the group of birational automorphisms of a nodal Q-factorial double cover $X$ of a smooth three-dimensional quadric branched over a quartic section. We also prove that $X$ is Q-factorial provided…

We prove that a general three-dimensional quartic $V$ in the complex projective space ${\mathbb P}^4$, the only singularity of which is a double point of rank 3, is a birationally rigid variety. Its group of birational self-maps is, up to…

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…

We introduce horizontal and vertical motivic invariants of birational maps between rational dominant maps and study their basic properties. As a first application, we show that the (usual) motivic invariants vanish for birational…

We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We…

This is the unabridged web version of the paper that will be published on the American Journal of Mathematics. In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is an…

Since the end of the XIXth century, we know that each birational map of the complex projective plane is the product of a finite number of quadratic birational maps of the projective plane; this motivates our work which essentially deals…

In this article we present a 3-dimensional analogue of a well-known theorem of E. Bombieri (in 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let $X$ be a projective minimal 3-fold of general type with…

We characterize the birational geometry of some hyperk\"ahler fourfolds of Picard rank $3$ obtained as the Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls. In each of the two cases considered, we identify all of…

For a geometrically rational surface X over an arbitrary field of characteristic different from 2 and 3 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of…

We classify pairs $(X,G)$ consisting of a (possibly singular) cubic threefold $X\subset\mathbb{P}^4$ and a finite subgroup $G\subset\mathrm{Aut}(X)$ such that $X$ is $G$-birationally rigid, i.e., $X$ is a $G$-Mori fiber space (over a…

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…

It is proved that a general Fano hypersurface of index 1 (in the projective space) with isolated singularities of general position is birationally rigid. Therefore it cannot be fibered into uniruled varieties of a smaller dimension by a…

We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16. The latter embeds naturally in the even…

Let f : X -> S be any elliptic fibration. If X has dimension 3 and is not uniruled, then X has a minimal model (with terminal singularities) [Mori]. In earlier work we have shown that there exists a birationally equivalent elliptic…

We classify elliptic fibrations birational to a nonsingular, minimal cubic surface over a field of characteristic zero. Our proof is adapted to provide computational techniques for the analysis of such fibrations, and we describe an…

Mutations of the cluster variables generating the cluster algebra of type $A^{(2)}_2$ reduce to a two-dimensional discrete integrable system given by a quartic birational map. The invariant curve of the map is a singular quartic curve, and…

A general linear determinantal quartic in $\mathbb{P}^4$ is nodal, non-$\mathbb{Q}$-factorial and rational. We show that the family $\mathcal{F}$ of such quartics also contains rational $\mathbb{Q}$-factorial quartics, and that a generic…

We prove that a finite group acting by birational automorphisms of a non-trivial Severi-Brauer surface over a field of characteristic zero contains a normal abelian subgroup of index at most 3. Also, we find an explicit bound for orders of…