arXiv subjects

Ronald van Luijk

Publications and source records attributed to Ronald van Luijk.

23 records · Page 2Linked to original sources

arXiv2006

Canonical vector heights on K3 surfaces with Picard number three -- addendum

In an earlier paper by the first author, an argument for the nonexistence of canonical vector heights on K3 surfaces of Picard number three was given, based on an explicit surface that was not proved to have Picard number three. In this paper, we fill the gap in the argument by redoing the computations for another explicit surface for which we prove that the Picard number equals three. The conclusion remains unchanged.

math.AG↗

arXiv2005

Quartic K3 surfaces without nontrivial automorphisms

For any field k of characteristic at most 5 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher characteristics. Over the rationals we construct an example on which the set of rational points is Zariski dense.

math.AG↗

arXiv2005

K3 surfaces with Picard number one and infinitely many rational points

In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Neron-Severi group over an algebraic closure of the base field, is high enough, more structure is known and more can be said. However, until recently not a single K3 surface was known to have geometric Picard number one. We give explicit examples of such surfaces over the rational numbers. This solves an old problem that has been attributed to Mumford. The examples we give also contain infinitely many rational points, thereby answering a question of Swinnerton-Dyer and Poonen.

math.AG↗

arXiv2005

A K3 surface associated to certain integral matrices with integral eigenvalues

In this article we will show that there are infinitely many symmetric, integral 3 x 3 matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire Neron-Severi group of this surface and find all low degree curves on it.

math.AG↗

arXiv2004

An elliptic K3 surface associated to Heron triangles

A rational triangle is a triangle with rational sides and rational area. A Heron triangle is a triangle with integral sides and integral area. In this article we will show that there exist infinitely many rational parametrizations, in terms of s, of rational triangles with perimeter 2s(s+1) and area s(s^2-1). As a corollary, there exist arbitrarily many Heron triangles with all the same area and the same perimeter. The proof uses an elliptic K3 surface Y. Its Picard number is computed to be 18 after we prove that the Neron-Severi group of Y injects naturally into the Neron-Severi group of the reduction of Y at a prime of good reduction. We also give some constructions of elliptic surfaces and prove that under mild conditions a cubic surface in projective three-space can be given the structure of an elliptic surface by cutting it with the family of hyperplanes through a given line L. Some of these constructions were already known but appear to have lacked proof in the literature until now.

math.AG↗