Related papers: On certain uniformity properties of curves over fu…

A smooth geometrically connected curve over the finite field $\mathbb{F}_q$ with gonality $\gamma$ has at most ${\gamma(q+1)}$ rational points. The first author and Grantham conjectured that there exist curves of every sufficiently large…

We establish the theorems that give necessary and sufficient conditions for an arbitrary function defined in the unit disk of complex plane in order to has boundary values along classes of equivalencies of simple curves. Our results…

Consider the projection of a smooth irreducible surface in $\mathbb{P}^3$ from a point. The uniform position principle implies that the monodromy group of such a projection from a general point in $\mathbb{P}^3$ is the whole symmetric…

This is a survey on recent results on counting of curves over finite fields. It reviews various results on the maximum number of points on a curve of genus g over a finite field of cardinality q, but the main emphasis is on results on the…

Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many…

The gonality of a smooth geometrically connected curve over a field $k$ is the smallest degree of a nonconstant $k$-morphism from the curve to the projective line. In general, the gonality of a curve of genus $g \ge 2$ is at most $2g - 2$.…

The curve graphs are not locally finite. In this paper, we show that the curve graphs satisfy a property which is equivalent to graphs being uniformly locally finite via Masur--Minsky's subsurface projections. As a direct application of…

We describe an algorithm to compute the number of points over finite fields on a broad class of modular curves: we consider quotients $X_H/W$ for $H$ a subgroup of $\GL_2(\mathbb Z/n\mathbb Z)$ such that for each prime $p$ dividing $n$, the…

The aim of this paper is to show how a conjectural lower bound on the canonical height function in the spirit of Lang and Silverman leads to an explicit uniform bound on the number of rational points on curves of genus $g\geq 2$ over a…

In this paper, we proved two results regarding the arithmetics of separably $\mathbb{A}^1$-connected varieties of rank one. First we proved over a large field, there is an $\mathbb{A}^1$-curve through any rational point of the boundary, if…

Let k be a field. We show that all homogeneous noncommutative curves of genus zero over k are noncommutative P^1-bundles over a (possibly) noncommutative base. Using this result, we compute complete isomorphism invariants of homogeneous…

Let $k$ be a field. Let $X/k$ be a stable curve whose geometric irreducible components are smooth rational curves. Taking Stein factorization of its normalization, we get a conic. We show the conic is non-split in certain cases. As an…

We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and for…

In this note, we give a new proof of Voisin's theorem on Green's conjecture for generic curves of odd genus resembling the first two sections of "Universal Secant Bundles and Syzygies of Canonical Curves" by the author, and so avoiding the…

We prove that on a closed surface of genus $g$, the cardinality of a set of simple closed curves in which any two are non-homotopic and intersect at most once is $\lesssim g^2 \log(g)$. This bound matches the largest known constructions to…

Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show…

We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures…

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method…

It is studied the Cauchy problem for the equations of Burgers' type but with bounded dissipation flux. Such equations degenerate to hyperbolic ones as the velocity gradient tends to infinity. Thus the discontinuous solutions are permitted.…

We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of…