The arithmetic of certain del Pezzo surfaces and \(K3\) surfaces (Q713182)

It is a classical result of Hasse that smooth quartic hypersurfaces always satisfy the Hasse principle. A degree 4 del Pezzo surface is the smooth intersection of two quadratic hypersurfaces of dimension 4. The author constructs degree 4 del Pezzo surfaces for which the Hasse principle fails because of the Brauer-Manin obstruction. They give explicit equations for the two quadric hypersurfaces defining the surface. One of the surfaces provided has been studied by \textit{B. J. Birch} and \textit{H. P. F. Swinnerton-Dyer} [J. Reine Angew. Math. 274--275, 164--174 (1975; Zbl 0326.14007)]. The author then uses the degree 4 del Pezzo surfaces to construct a family of \(K3\) surfaces violating the Hasse principle also. These surfaces are double covers of the del Pezzo branched over a smooth conic.