Related papers: Counting rational points on quartic del Pezzo surf…

This article focuses on the study of toric algebraic statistical models which correspond to toric Del Pezzo surfaces with Du Val singularities. A closed-form for the Maximum Likelihood Estimate of algebraic statistical models which…

Some classes of cubic fourfolds are birational to fibrations over $P^2$, where the fibers are rational surfaces. This is the case for cubics containing a plane (resp. an elliptic ruled surface), where the fibers are quadric surfaces (resp.…

We classify geometrically integral regular del Pezzo surfaces which are not geometrically normal over imperfect fields of positive characteristic. Based on this classification, we show that a three-dimensional terminal del Pezzo fibration…

We introduce the split torsor method to count rational points of bounded height on Fano varieties. As an application, we prove Manin's conjecture for all nonsplit quartic del Pezzo surfaces of type $\mathbf A_3+\mathbf A_1$ over arbitrary…

This article wants to show two things: first, that certain problems in Diophantus' Arithmetica lead to equations defining del Pezzo surfaces or other rational surfaces, while certain others lead to K3 surfaces; second, that Diophantus' own…

We give a computable lower bound on the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height bounds on components of the Noether--Lefschetz…

For an arbitrary del Pezzo surface S, we compute alpha(S), which is the volume of a certain polytope in the dual of the effective cone of S, using Magma and Polymake. The constant alpha(S) appears in Peyre's conjecture for the leading term…

Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a…

Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface,…

The number of rational points of a plane non-singular algebraic curve X defined over a finite field is computed, provided that the generic point of X is not an inflexion and that X is Frobenius non-classical with respect to conics.

In this paper, we give an effective and efficient algorithm which on input takes non-zero integers $A$ and $B$ and on output produces the generators of the Mordell-Weil group of the elliptic curve over $\mathbb{Q}(t)$ given by an equation…

In this paper, we will give a uniform upper bound of the number of rational points of bounded height in non-singular curves by applying the global determinant method.

For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined…

Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms…

Let $k$ be an infinite field of characteristic 0, and $X$ a del Pezzo surface of degree $d$ with at least one $k$-rational point. Various methods from algebraic geometry and arithmetic statistics have shown the Zariski density of the set…

We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type $\left\{k_i\times\frac{1}{r_i}(1,a_i): 3\le r_i \le 10,k_i \in \ZZ_{\ge 0}\right\}$; as well-formed and quasismooth varieties…

We find upper and lower bounds on the number of rational points that are $\psi$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle…

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses…

In this note we construct several infinite families of diagonal quartic surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where $a,b,c,d\in\Z\setminus\{0\}$ with infinitely many rational points and satisfying the condition…

We establish the sharp estimate <<_d N^{2/d} for the number of rational points of height at most N on an irreducible projective curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients…