On integral points on open degree four del Pezzo surfaces

ALGEBRAIC GEOMETRY & NUMBER THEORY SEMINAR

I will report on investigations, joint with Damaris Schindler (Göttingen), concerning the 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 $\bbQ$ and over $\bbZ_p$ for all primes $p$, but not over $\bbZ$. By blow-up, these yield cubic polynomials in three variables all integral solutions of which satisfy a gcd condition.