Damaris Schindler

On integral points on degree four del Pezzo surfaces

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 Q and over Zp 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.

Del Pezzo surfaces of degree four violating the Hasse principle are Zariski dense in the moduli scheme@@@Les surfaces de del Pezzo de degré quatre qui violent le principe de Hasse sont Zariski-denses dans le schéma de modules

We show that, over every number field, the degree four del Pezzo surfaces that violate the Hasse principle are Zariski dense in the moduli scheme.