Related papers: Counting rational points on del Pezzo surfaces wit…

Given a nonsingular quartic del Pezzo surface, a conjecture of Manin predicts the density of rational points on the open subset of the surface formed by deleting the lines. We prove that this prediction is of the correct order of magnitude…

We give a correspondence which associates, to any Del Pezzo surface X of degree 6 over a field k of characteristic 0, a collection of data consisting of a Severi-Brauer variety/k and a set of points defined over some extension of k.

Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is…

Given an elliptic surface $\mathcal{E}\to\mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where…

In this paper, we prove that a pair of the minimal resolution of a del Pezzo surface with rational double points whose general anti-canonical member is smooth and its exceptional divisor lifts to the Witt ring. We also classify a del Pezzo…

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.

We prove that any numerically exceptional collection of maximal length, consisting of line bundles, on a smooth del Pezzo surface is a standard augmentation in the sense of L.Hille and M.Perling. We deduce that any such collection is…

Detailed illustration of the method for calculating the Chow group of a rational surface over a local field [math.AG/0302157 (th.~4)], applied to a certain del Pezzo surface of degree~4. Involves the construction of a regular integral model…

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…

We explicitly construct a component of the K-moduli space of K-polystable del Pezzo surfaces which is a smooth rational curve.

The parametric degree of a rational surface is the degree of the polynomials in the smallest possible proper parametrization. An example shows that the parametric degree is not a geometric but an arithmetic concept, in the sense that it…

We consider some families of smooth Fano hypersurfaces $X_{n+2}$ in ${\bf P}^{n+2} \times {\bf P}^3$ given by a homogeneous polynomial of bidegree $(1,3)$. For these varieties we obtain lower bounds for the number of $F$-rational points of…

Previous work of the authors showed that every quartic del Pezzo surface over a number field has index dividing $2$ (i.e., has a closed point of degree $2$ modulo $4$),, and asked whether such surfaces always have a closed point of degree…

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.

We construct the first examples of regular del Pezzo surfaces for which the irregularity (i.e. the dimension of the first cohomology group of the structure sheaf) is nonzero. We also find a restriction on the integer pairs that are possible…

The surface $z^2=ay^2+P(x), \, a \in k, \, P(x) \in k[x]$ is not $k$-rational, if $a \not\in k^2$ and $P(x)$ satisfies some conditions. This result essentially due to Iskovskih but his statement is in terms of algebraic geometry, and not so…

We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When $p=2$, this…

In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect…

We construct an algorithm for solving the following problem: given a number field $K$, a positive integer $N$, and a positive real number $B$, determine all points in $\mathbb P^N(K)$ having relative height at most $B$. A theoretical…

We consider diagonal cubic surfaces defined by an equation of the form ax^3+by^3+cz^3+dt^3 = 0. Numerically, one can find all rational points of height < B for B in the range of up to 100 000, thanks to a program due to D. J. Bernstein. On…