28 research outputs found
Cohomology and the Brauer groups of diagonal surfaces
We present a method for calculating the Brauer group of a surface given by a
diagonal equation in the projective space. For diagonal quartic surfaces with
coefficients in Q we determine the Brauer groups over Q and Q(i).Comment: 45 page
Effectivity of Brauer-Manin obstructions on surfaces
We study Brauer-Manin obstructions to the Hasse principle and to weak
approximation on algebraic surfaces over number fields.Comment: 13 page
On the Brauer group of diagonal quartic surfaces
We obtain an easy sufficient condition for the Brauer group of a diagonal
quartic surface D over Q to be algebraic. We also give an upper bound for the
order of the quotient of the Brauer group of D by the image of the Brauer group
of Q. The proof is based on the isomorphism of the Fermat quartic surface with
a Kummer surface due to Masumi Mizukami.Comment: Appendix "The Fermat quartic as a Kummer surface (after Mizukami)",
by Sir Peter Swinnerton-Dye
Transcendental obstructions to weak approximation on general K3 surfaces
We construct an explicit K3 surface over the field of rational numbers that
has geometric Picard rank one, and for which there is a transcendental
Brauer-Manin obstruction to weak approximation. To do so, we exploit the
relationship between polarized K3 surfaces endowed with particular kinds of
Brauer classes and cubic fourfolds.Comment: 24 pages, 3 figures, Magma scripts included at the end of the source
file