In 1970, Manin showed that the Brauer group can obstruct the existence of rational points. Colliot-Thélène and Sansuc have conjectured that this obstruction completely explains the failure of rational points on del Pezzo surfaces. We show that on degree 4 del Pezzo surfaces, this Brauer-Manin obstruction manifests itself through linear projections. As a consequence of the proof, we obtain a simple and efficient for computing the Brauer classes of a degree 4 del Pezzo surface. This is joint work with Anthony Várilly-Alvarado.
Obstructions to the Hasse principle on degree 4 del Pezzo surfaces
Nov. 18, 2014 12:10pm (MATH …
Kempner
Robert Varley (University of Georgia)
X
Rational homotopy theory as developed by Quillen and Sullivan allows a complete description of its objects of study by differential graded algebras over the rational numbers. Thus rationally some of the basic classification problems in homotopy theory become "moduli problems" in algebraic geometry and number theory. Halperin, Schlessinger, and Stasheff showed how to describe all rational homotopy types having a given cohomology algebra. I will discuss this moduli construction and how completion of the rationals is useful for its analysis. In particular, homotopy theory over the finite adeles seems to accommodate the known fact that even for smooth projective varieties defined over a number field, the rational homotopy type of the associated topological space can depend on the embedding of the number field in the complex numbers.
I will recount some of the interesting features of principally polarized abelian varieties and their theta divisors, emphasizing the case of Jacobians of curves, connections with linear series, and the intensive study made by Kempf of the presentation matrix for the determinantal structure of Jacobian theta divisors. The analogous setup for Pryms is more complicated. Roy Smith and I have been trying to supplement Casalaina-Martin's multiplicity formula by using a natural geometric parametrization of the tangent cone. I will describe the skew-symmetric presentation matrix for the (local) Pfaffian structure of Prym theta divisors. It is then clear which essential higher order information is still lacking for very exceptional singular points, and I will explain how the unknown 2nd order terms seem to be governed by a Massey product structure.
Obstructions to the Hasse principle on degree 4 del Pezzo surfaces