Yuri Tschinkel

Rational points of bounded height on compactifications of anisotropic tori

We investigate the analytic properties of the zeta-function associated with heights on equivariant compactifications of anisotropic tori over number fields. This allows to verify conjectures about the distribution of rational points of bounded height.

Weak approximation over function fields

We prove that rationally connected varieties over the function field of a complex curve satisfy weak approximation for places of good reduction.

Height

We investigate analytic properties of height zeta functions of toric varieties. Using the height zeta functions, we prove an asymptotic formula for the number of rational points of bounded height with respect to an arbitrary line bundle whose first Chern class is contained in the interior of the cone of effective divisors

\zeta

— We study the equations of universal torsors on rational surfaces.

functions of toric varieties

We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.

Universal Torsors and Cox Rings

We develop a mixed-characteristic version of the MoriMukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K3 surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.

IGUSA INTEGRALS AND VOLUME ASYMPTOTICS IN ANALYTIC AND ADELIC GEOMETRY

Given a variety over a number field, are its rational points potentially dense, i.e. does there exist a finite extension over which rational points are Zariski dense? We study the question of potential density for symmetric products of surfaces. Contrary to the situation for curves, rational points are not necessarily potentially dense on a sufficiently high symmetric product. Our main result is that rational points are potentially dense for the Nth symmetric product of a K3 surface, where N is explicitly determined by the geometry of the surface. The basic construction is that for some N, the Nth symmetric power of a K3 surface is birational to an Abelian fibration over ℙN. It is an interesting geometric problem to find the smallest N with this property.

Constructing rational curves on K3 surfaces

A refined version of Maninsconjecture aboutthe asymptotics of points of bounded height on Fano varieties has been developped by Batyrev and the authors. We test numerically this refined conjecture for some diagonal cubic surfaces.

Abelian fibrations and rational points on symmetric products

We discuss Manins conjecture (with Peyres refinement) concerning the distrib- ution of rational points of bounded height on Del Pezzo surfaces, by highlighting the use of universal torsors in such counting problems. To illustrate the method, we provide a proof of Manins conjecture for the unique split singular quartic Del Pezzo surface with a singularity of type D4.

Tamagawa numbers of diagonal cubic surfaces, numerical evidence

We study the arithmetic of certain del Pezzo surfaces of degree 2. We produce examples of Brauer-Manin obstruction to the Hasse principle, coming from 2- and 4-torsion elements in the Brauer group.