Algebraic Geometry Seminar

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On the dependence of the Brauer-Manin obstruction on the degree of a variety

Speaker: Bianca Viray, University of Washington

Location: Warren Weaver Hall 317

Date: Tuesday, October 24, 2017, 3:30 p.m.

Synopsis:

Let X be a smooth projective variety of degree d over a number field k.  In 1970 Manin observed that elements of the Brauer group of X can obstruct the existence of a k-point, even when X is everywhere locally soluble.  In joint work with Brendan Creutz, we prove that if X is geometrically abelian, Kummer, or bielliptic then this Brauer-Manin obstruction to the existence of a k-point can be detected from only the d-primary torsion Brauer classes.  (In the case when X is bielliptic, the result is conditional on the finiteness of the Tate-Shafarevich group for j-invariant 0 elliptic curves.). If time permits, we will discuss ongoing joint work with Creutz and Voloch that shows that the situation is quite different for general type curves.