Remarks on the Milnor conjecture over schemes

The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging from sums of squares to the structure of absolute Galois groups. Here, we survey some recent work on generalizations of the Milnor conjecture to the context of schemes (mostly smooth varieties over fields of characteristic not 2). Surprisingly, a version of the Milnor conjecture fails to hold for certain smooth complete p-adic curves with no rational theta characteristic (this is the work of Parimala, Scharlau, and Sridharan). We explain how these examples fit into the larger context of an unramified Milnor question, offer a new approach to the question, and discuss new results in the case of curves over local fields and surfaces over finite fields.Comment: 23 page

Failure of the local-global principle for isotropy of quadratic forms over rational function fields

We prove the failure of the local-global principle, with respect to all discrete valuations, for isotropy of quadratic forms over a rational function field of transcendence degree at least 2 over the complex numbers. Our construction involves the generalized Kummer varieties considered by Borcea and Cynk--Hulek.Comment: 7 pages, comments welcome

Stable rationality of quadric and cubic surface bundle fourfolds

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2,3) in P^2 x P^3 is not stably rational. Via projections onto the two factors, X is a cubic surface bundle over P^2 and a conic bundle over P^3, and we analyze the stable rationality problem from both these points of view. This provides another example of a smooth family of rationally connected fourfolds with rational and nonrational fibers. Finally, we introduce new quadric surface bundle fourfolds over P^2 with discriminant curve of any even degree at least 8, having nontrivial unramified Brauer group and admitting a universally CH_0-trivial resolution.Comment: 27 pages, comments welcome

Stable rationality of quadric and cubic surface bundle fourfolds

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2,3) in P^2 x P^3 is not stably rational. Via projections onto the two factors, X is a cubic surface bundle over P^2 and a conic bundle over P^3, and we analyze the stable rationality problem from both these points of view. This provides another example of a smooth family of rationally connected fourfolds with rational and nonrational fibers. Finally, we introduce new quadric surface bundle fourfolds over P^2 with discriminant curve of any even degree at least 8, having nontrivial unramified Brauer group and admitting a universally CH_0-trivial resolution.Comment: 27 pages, comments welcome