Construction of a del Pezzo surface with maximal Galois action on its Picard group

Abstract

A del Pezzo surface of degree d over a field K is a smooth projective surface X over K such that there exists a finite field extension over which X is isomorphic to P 2 blown up in 9 — d rational points in “general” position. In this paper we give a detailed construction of a del Pezzo surface of degree 2 over Q having maximal Galois action on Pic(X Q ̄ ) . We follow the method given by Ekedahl in [3] where, as an application of techniques used to prove the main result of that article, he constructs a del Pezzo surface of degree 3 over Q (a cubic surface) having this property.