Related papers: Curves on Frobenius classical surfaces in $\mathbb…

A plane curve $C$ in $\mathbb{P}^2$ defined over $\mathbb{F}_q$ is called plane-filling if $C$ contains every $\mathbb{F}_q$-point of $\mathbb{P}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth…

Nonsingular plane curves over a finite field $\mathbb{F}_q$ of degree $q+2$ passing through all the $\mathbb{F}_q$-points of the plane admita representation by $3\times 3$ matrices over $\mathbb{F}_q$. We classify their degenerations by…

Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic)…

In \cite{MP} we have shown that if a compact Riemann surface admits a Strebel differential with rational periods, then the Riemann surface is the complex model of an algebraic curve defined over the field of algebraic numbers. We will show…

We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that…

We determine in this paper the distribution of the number of points on the cyclic covers of $\mathbb{P}^1(\mathbb{F}_q)$ with affine models $C: Y^r = F(X)$, where $F(X) \in \mathbb{F}_q[X]$ and $r^{th}$-power free when $q$ is fixed and the…

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously…

Consider a smooth, geometrically irreducible, projective curve of genus $g \ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that…

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a…

Let X be the graph in the plane of a pfaffian function f (in the sense of Khovanskii). Suppose X is not algebraic. This note gives an upper bound for the number of rational points on X of height up to X. The bound is uniform in the order…

We give an asymptotic lower bound on the number of field extensions generated by algebraic points on superelliptic curves over $\mathbb{Q}$ with fixed degree $n$ and discriminant bounded by $X$. For $C$ a fixed such curve given by an affine…

In this paper we compute the number of rational curves with one node passing through a given number of points, lines and tangent to a given number of planes in $\mathbb{P}^3$.

Let $\mathbb{F}_q$ be a finite field with $q=p^n$ elements. In this paper, we study the number of $\mathbb{F}_q$-rational points on the affine hypersurface $\mathcal X$ given by $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$, where…

A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first…

Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…

Consider a Mordell curve $E_a:y^2=x^3+a$ with $a \in \mathbb Z$. These curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on the rank of the $\varphi$-Selmer group of $E_a$ over $\mathbb Q(\zeta_3)$ in…

We use class field theory to search for curves with many rational points over small finite fields. By going through abelian covers of curves of small genus we find a number of new curves. In particular, we settle the question of how many…

In this paper, we give a uniform upper bound on the rational points of bounded height provided by conics in a cubic surface. For this target, we give a generalized version of the global determinant method of Salberger by Arakelov geometry.

Consider a finite scheme of length l contained in a smooth quadric surface over the complex numbers. We determine the number of linearly independent curves passing through the scheme, of degree at least l - 1.

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…