Related papers: $\mathbb{A}^1$-connected varieties over non-closed…

We construct isotrivial and non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type…

We study the asymptotic growth of the number of rational points of bounded height on smooth projective split toric varieties with Picard rank 2 over number fields, with respect to Arakelov height functions associated with big metrized line…

We show that if a family of complex varieties over a base B admits a section when restricted to a very general curve in B, then the family must contain a subfamily of rationally connected varieties dominating B. As an application, we deduce…

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…

Given a morphism between smooth projective varieties $f: W \to X$, we study whether $f$-relatively free rational curves imply the existence of $f$-relatively very free rational curves. The answer is shown to be positive when the fibers of…

Let $A_1, A_2\in \mathbb C(z)$ be rational functions of degree at least two that are neither Latt\`es maps nor conjugate to $z^{\pm n}$ or $\pm T_n.$ We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of…

Let X be a smooth, projective variety defined over a local field K. Following Manin, two K-points of X are called R-equivalent if they can be joined by a rational curve defined over K. The main result of this note shows that if there are…

A theorem of B. Green states that if A is a Dedekind ring whose fraction field is a local or global field, every normal projective curve over Spec(A) has a finite morphism to P^1_A. We give a different proof of a variant of this result…

Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of…

We prove that every curve on a rationally connected variety is algebraically equivalent to a (non-effective) integral sum of rational curves.

According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that…

We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first example, to our knowledge, of a simply connected smooth variety whose sets of integral points are never…

This paper provides a non-standard analogue of Bezout's theorem. This is acheived by showing that, in all characteristics, the notion of Zariski multiplicity coincides with intersection multiplicity when we consider the full families of…

We prove a few uniform versions of the Mordell-Lang Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points. The main focus is on function fields having high transcendence degree over the…

Let $X$ be a projective variety and let $C$ be a rational normal curve on $X$. We compute the normal bundle of $C$ in a general complete intersection of hypersurfaces of sufficiently large degree in $X$. As a result, we establish the…

In [3], it is proved that the quotient of an abelian variety $A$ by a finite order automorphism $g$ is uniruled if and only if some power of $g$ satisfies a numerical condition $0<\age(g^k)<1$. In this paper, we show that $\age(g^k)=1$ is…

We consider the potential density of rational points on an algebraic variety defined over a number field $K$, i.e., the property that the set of rational points of $X$ becomes Zariski dense after a finite field extension of $K$. For a…

Let X be a smooth hypersurface of degree d in P^n over an algebraically closed field of characteristic p. We show that X must be separably rationally connected and must contain a free line if either p is at least d or if p is at least d-1…

We show that the sheaf of $\mathbb A^1$-connected components of a Nisnevich sheaf of sets and its universal $\mathbb A^1$-invariant quotient (obtained by iterating the $\mathbb A^1$-chain connected components construction and taking the…

In this paper, the technique of foliations in characteristic $p$ is used to investigate the difference between rational connectedness and separable rational connectedness in positive characteristic. The notion of being freely rationally…