Related papers: G\'eom\'etrie, points entiers et courbes enti\`ere…

For a projective variety X defined over a field K, there is a special class of self-morphisms of X called algebraic dynamical systems. In this paper we take K to be the function field of a smooth curve and prove that at each place of K,…

We explore an analogue of the Andr\'e-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety $X$ of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM)…

We prove that generic Hitchin representations are strongly dense: every pair of non commuting elements in their image generate a Zariski-dense subgroup of SL_n(R). The proof uses a theorem of Rapinchuk, Benyash-Krivetz and Chernousov, to…

We investigate some general questions in algebraic dynamics in the case of generic endomorphisms of projective spaces over a field of characteristic zero. The main results that we prove are that a generic endomorphism has no non-trivial…

This is Part II of a series of three papers. We studies the hyperbolicity of complex quasi-projective varieties $X$ in the presence of a big and reductive representation $\varrho: \pi_1(X)\to {\rm GL}_N(\mathbb{C})$. For any Galois…

Let $K$ be a Birch field, that is, a field for which every diagonal form of odd degree in sufficiently many variables admits a non-zero solution; for example, $K$ could be the field of rational numbers. Let $f_1, \ldots, f_r$ be homogeneous…

In this note we consider the problem of integrality of Zariski decompositions for pseudoeffective integral divisors on algebraic surfaces. We show that while sometimes integrality of Zariski decompositions forces all negative curves to be…

For a reductive group $G$ over a finite field $k$, and a smooth projective curve $X/k$, we give a motivic counting formula for the number of absolutely indecomposable $G$-bundles on $X$. We prove that the counting can be expressed via the…

We classify invariant probability measures for non-elementary groups of automorphisms, on any compact K\"ahler surface X, under the assumption that the group contains a so-called "parabolic automorphism". We also prove that except in…

Working over a base number field $\KK$, we study the attractive question of Zariski non-density for $(D,S)$-integral points in $\mathrm{O}_f(x)$ the forward $f$-orbit of a rational point $x \in X(\KK)$. Here, $f \colon X \rightarrow X$ is a…

In this paper, we provide evidence to support a positive answer to a question of Hassett and Tschinkel. In particular, if an algebraic variety V has a dense set of rational points, they ask whether or not the set of D-integral points is…

Let $k$ be a number field and $U$ a smooth integral $k$-variety. Let $X \to U$ be an abelian scheme. We consider the set $\mathcal{R}$ of rational points $m \in U(k)$ such that the Mordell-Weil rank of the fibre $U_{m}$ is strictly bigger…

Let $G$ be a real, reductive algebraic group, and let $X$ be a homogeneous space for $G$ with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of…

Let $K$ be a number field. For positive integers $m$ and $n$ such that $m\mid n$, we let $\mathscr{S}_{m,n}$ be the set of elliptic curves $E/K$ defined over $K$ such that $E(K)_{\operatorname{tors}}\supseteq \mathscr{T}\cong…

Let X be a normal projective variety admitting an action of a semisimple group with a unique closed orbit. We construct finitely many rational curves in X, all having a common point, such that every effective one-cycle on X is rationally…

We show that given a smooth projective variety X over C with dim(X) > 2, an ample line bundle O(1) on X and an integer n > 1, any n distinct points on a generic hypersurface of degree d in X are linearly independent in CH_0(X) if d >> 0.…

In this note, we propose a geometric analogue of Dirichlet's unit theorem on arithmetic varieties, that is, if X is a normal projective variety over a finite field and D is a pseudo-effective Q-Cartier divisor on X, does it follow that D is…

Let Y be a complex algebraic curve and let [Y]={X_1,...,X_n} be the set of all real algebraic curves X_i with complexification X_i(C)=Y, such that the real points X_i(R) divide X_i(C). We find all such families [Y]. According to Harnak…

Let $X$ be a smooth irreducible projective variety of dimension at least 2 over an algebraically closed field of characteristic 0 in the projective space ${\mathbb{P}}^n$. Bertini's Theorem states that a general hyperplane $H$ intersects…

Given a variation of Hodge structures on a quasi-projective base $S$, whose generic Mumford-Tate group is non-product, we prove that the (countable) union of positive components of the Hodge locus is either an algebraic subvariety of $S$,…