32 research outputs found
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
Brill-Noether special cubic fourfolds of discriminant 14
We study the Brill-Noether theory of curves on K3 surfaces that are Hodge
theoretically associated to cubic fourfolds of discriminant 14. We prove that
any smooth curve in the polarization class has maximal Clifford index and
deduce that a cubic fourfold contains disjoint planes if and only if it admits
a Brill-Noether special associated K3 surface of degree 14. As an application,
the complement of the pfaffian locus, inside the Noether-Lefschetz divisor of
discriminant 14 in the moduli space of cubic fourfolds, is contained in the
irreducible locus of cubic fourfolds containing two disjoint planes.Comment: 19 page
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