On integral points on degree four del Pezzo surfaces

TL;DR

This paper investigates algebraic and transcendental Brauer-Manin obstructions affecting integral points on degree four del Pezzo surfaces, providing concrete examples and exploring archimedean obstructions.

Contribution

It introduces new examples illustrating obstructions to integral points on del Pezzo surfaces and discusses archimedean obstructions in this context.

Findings

Concrete examples of quadratic polynomials with local solutions but no integral solutions.

Analysis of Brauer-Manin obstructions at archimedean places.

Connections between geometric obstructions and gcd conditions in solutions.

Abstract

We report on our investigations concerning algebraic and transcendental Brauer-Manin obstructions to integral points on complements of a hyperplane section in degree four del Pezzo surfaces. We discuss moreover two concepts of an obstruction at an archimedean place. Concrete examples are given of pairs of non-homogeneous quadratic polynomials in four variables representing (0,0) over Q and over Z_p for all primes p, but not over Z. By blow-up, these yield cubic polynomials in three variables all integral solutions of which satisfy a gcd condition.