On the Arithmetic of Del Pezzo Surfaces of Degree 2

Del Pezzo surfaces are smooth projective surfaces, isomorphic over the algebraic closure of the base ,eld to P P or the blow-up of P in up to eight points in general position. In the latter case the del Pezzo surface has degree equal to 9 minus the number of points in the blow-up. The arithmetic of del Pezzo surfaces over number ,elds is an active area of investigation. It is known that the Hasse principle holds for del Pezzo surfaces of degree at least 5. Counterexamples to the Hasse principle were discovered for del Pezzo surfaces of degrees 3 and 4 (see [17] and [1], respectively). A growing body of evidence (for instance, [5]) led to the question of whether the failure of the Hasse principle for del Pezzo surfaces is always explained by theBrauer--Manin obstruction; this question is speci,cally raised by Colliot-Th@elAene and Sansuc in [7]. Computer veri,cations for diagonal cubics in [6] and theoretical advances, such as [4, 14, 20], lend support to an aBrmative answer to this question. A del Pezzo surface of degree 2 can be realised as a double cover of P rami,ed in a smooth quartic curve. In this note we consider surfaces S over Q of the form