72 research outputs found
More cubic surfaces violating the Hasse principle
We generalize L.J. Mordell's construction of cubic surfaces for which the
Hasse principle fails
The Hasse principle for lines on diagonal surfaces
Given a number field and a positive integer , in this paper we
consider the following question: does there exist a smooth diagonal surface of
degree in over which contains a line over every
completion of , yet no line over ? We answer the problem using Galois
cohomology, and count the number of counter-examples using a result of
Erd\H{o}s.Comment: 14 page
On the Brauer-Manin obstruction for degree four del Pezzo surfaces
We show that, for every integer and every finite set of
places, there exists a degree del Pezzo surface over such
that and the
Brauer-Manin obstruction works exactly at the places in . For , we
prove that in all cases, with the exception of , this surface
may be chosen diagonalizably over
On the frequency of algebraic Brauer classes on certain log K3 surfaces
Given systems of two (inhomogeneous) quadratic equations in four variables,
it is known that the Hasse principle for integral points may fail. Sometimes
this failure can be explained by some integral Brauer-Manin obstruction. We
study the existence of a non-trivial algebraic part of the Brauer group for a
family of such systems and show that the failure of the integral Hasse
principle due to an algebraic Brauer-Manin obstruction is rare, as for a
generic choice of a system the algebraic part of the Brauer-group is trivial.
We use resolvent constructions to give quantitative upper bounds on the number
of exceptions.Comment: 13 page
On the number of certain Del Pezzo surfaces of degree four violating the Hasse principle
We give an asymptotic expansion for the density of del Pezzo surfaces of
degree four in a certain Birch Swinnerton-Dyer family violating the Hasse
principle due to a Brauer-Manin obstruction. Under the assumption of Schinzel's
hypothesis and the finiteness of Tate-Shafarevich groups for elliptic curves,
we obtain an asymptotic formula for the number of all del Pezzo surfaces in the
family, which violate the Hasse principle.Comment: 27 page
On the algebraic Brauer classes on open degree four del Pezzo surfaces
We study the algebraic Brauer classes on open del Pezzo surfaces of degree
. I.e., on the complements of geometrically irreducible hyperplane sections
of del Pezzo surfaces of degree . We show that the -torsion part is
generated by classes of two different types. Moreover, there are two types of
-torsion classes. For each type, we discuss methods for the evaluation of
such a class at a rational point over a -adic field
On the quasi group of a cubic surface over a finite field
We construct nontrivial homomorphisms from the quasi group of some cubic
surfaces over \bbF_{\!p} into a group. We show experimentally that the
homomorphisms constructed are the only possible ones and that there are no
nontrivial homomorphisms in the other cases. Thereby, we follow the
classification of cubic surfaces, due to A. Cayley
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