On integral points on degree four del Pezzo surfaces

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 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 and over p for all primes p, but not over. By blow-up, these yield cubic polynomials in three variables all integral solutions of which satisfy a gcd condition.