Related papers: Rational points on singular intersections of quadr…

In this paper, we continue the study of the relation between rational points of rational elliptic surfaces and plane curves. As an application, we give first examples of Zariski pairs of cubic-line arrangements that do not involve…

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…

The topology of the intersection of three quadrics in Euclidean 6-space is studied using Kollar results. This needs an existence of a line without real points in the complex projectivisation of quadrics. We establish the existence of such a…

Let $f,g\in\mathbb{Z}[u_1,u_2]$ be binary quadratic forms. We provide upper bounds for the number of rational points $(u,v)\in\mathbb{P}^1(\mathbb{Q})\times\mathbb{P}^1(\mathbb{Q})$ such that the ternary conic \[ X_{(u,v)}: f(u_1,u_2)x^2 +…

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface $x_1y_1^2+\dots+x_4y_4^2=0$ in…

A sequence of rational points on an algebraic planar curve is said to form an $r$-geometric progression sequence if either the abscissae or the ordinates of these points form a geometric progression sequence with ratio $r$. In this work, we…

We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. We combine the methods and results from three different papers.

A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely…

We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E/\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \geq 1$, $d \geq 1$…

We estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope.

In this paper we consider the density of rational points on the "stacky" curve $\mathcal{X}(\mathbb{P}^1;0,2;1,2;\infty,2)$ which is $\mathbb{P}^1$ with three half points, with respect to the so-called Ellenberg-Satriano-Zuerick-Brown…

We show that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 12, if the minimal resolution of $X$ is not a…

Let $\mathcal{X}$ be a projective irreducible nonsingular algebraic curve defined over a finite field $\mathbb{F}_q$. This paper presents a variation of the St\"orh-Voloch theory and sets new bounds to the number of…

We consider transient nearest neighbor random walks on the positive part of the real line. We give criteria for the finiteness of the number of cutpoints and strong cutpoints. Examples and open problems are presented.

We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three by producing explicit rational parametrizations given by polynomials of low degree. As a byproduct of our…

We give elementary proof of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces $X\subset \mathbb{P}^n(\mathbb{R})$. The main tool is a refinement of the simplex lemma, which…

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…

Building on work of Segre and Koll'ar on cubic hypersurfaces, we construct over imperfect fields of characteristic p\geq 3 particular hypersurfaces of degree p, which show that geometrically rational schemes that are regular and whose…

In this paper,we count the rational points on the weighted projective spaces defined over number fields w.r.t. ``size''. An asymptotic formula which generalizes the result of Schanuel's ``Heights in number fields'' is obtained. Furthermore,…

We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.