Some remarks on morphisms between Fano threefolds

Let $X$, $Y$ be Fano threefolds of Picard number one and such that the ample generators of Picard groups are very ample. Let $X$ be of index one and $Y$ be of index two. It is shown that the only morphisms from $X$ to $Y$ are double coverings. In fact nearly the whole paper is the analysis of the case where $Y$ is the linear section of the Grassmannian G(1,4), since the other cases were more or less solved in another article. This remaining case is treated with the help of Debarre's connectedness theorem for inverse images of Schubert cycles.Comment: 14 pages, LaTeX. A lemma added in Section

On morphisms onto quadrics

It is proved that the degree of a morphism from a smooth projective n-fold with Picard number one to a smooth n-quadric is bounded (provided, of course, that n is at least three). Actually it has been proved some years ago, but I have never written down the proof, until J.-M. Hwang suggested, recently, that I do.Comment: 3 pages, LaTeX. A theorem claimed a long time ago but never written dow

Some applications of p-adic uniformization to algebraic dynamics

This is not a research paper, but a survey submitted to a proceedings volume.Comment: 21 pages, LaTe

A computation of invariants of a rational self-map

I compute the dynamical degrees in C. Voisin's example of a rational self-map of the variety of lines on a cubic fourfold.Comment: LaTeX, 10 page

A remark on a question of Beauville about lagrangian fibrations

This note is a proof of the fact that a lagrangian torus on a hyperkaehler fourfold is always a fiber of an almost holomorphic lagrangian fibration.Comment: LaTeX, 3 pages; minor changes following referee's repor

Existence of non-preperiodic algebraic points for a rational self-map of infinite order

Let $X$ be a variety defined over a number field and $f$ be a dominant rational self-map of $X$ of infinite order. We show that $X$ admits many algebraic points which are not preperiodic under $f$. If $f$ were regular and polarized, this would follow immediately from the theory of canonical heights, but it does not work very well for rational self-maps. We provide an elementary proof following an argument by Bell, Ghioca and Tucker (arxiv:0808.3266).Comment: 6 pages, LaTe

On an automorphism of Hilb[2]Hilb^{[2]} of certain K3 surfaces

An example of potential density of rational points on the second punctual Hilbert scheme of certain K3 surfaces is treated in detail. This is an amplification of some remarks made by O'Grady and Oguiso.Comment: 6 pages, LaTe

Potential density of rational points on the variety of lines of a cubic fourfold

We prove the potential density of rational points on the variety of lines of a sufficiently general cubic fourfold defined over a number field, where ``sufficiently general'' means that a condition of Terasoma type is satisfied. These varieties have trivial canonical bundle and have geometric Picard group equal to $\mathbb{Z}$.Comment: LaTeX, 22 page

Rational curves on hyperkahler manifolds

Let $M$ be an irreducible holomorphically symplectic manifold. We show that all faces of the Kahler cone of $M$ are hyperplanes $H_i$ orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kahler cone is a connected component of a complement of the positive cone to the union of all $H_i$. We provide several characterizations of the MBM-classes. We show the invariance of MBM property by deformations, as long as the class in question stays of type (1,1). For hyperkahler manifolds with Picard group generated by a negative class $z$, we prove that $\pm z$ is Q-effective if and only if it is an MBM class. We also prove some results towards the Morrison-Kawamata cone conjecture for hyperkahler manifolds.Comment: 34 page

Contraction centers in families of hyperkahler manifolds

We study the exceptional loci of birational (bimeromorphic) contractions of a hyperk\"ahler manifold $M$. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the K\"ahler cone. Homology classes which can possibly be orthogonal to a wall of the K\"ahler cone of some deformation of $M$ are called MBM classes. We prove that all MBM classes of type (1,1) can be represented by rational curves, called MBM curves. All MBM curves can be contracted on an appropriate birational model of $M$, unless $b_2(M) \leq 5$. When $b_2(M)>5$, this property can be used as an alternative definition of an MBM class and an MBM curve. Using the results of Bakker and Lehn, we prove that the diffeomorphism type of a contraction locus remains stable under all deformations for which these classes remains of type (1,1), unless the contracted variety has $b_2\leq 4$. Moreover, these diffeomorphisms preserve the MBM curves, and induce biholomorphic maps on the contraction fibers, if they are normal.Comment: 34 pages, version 1.4. Supersedes arXiv:1804.0046