Vol. 12, No. 2, 2018

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Quadric surface bundles over surfaces and stable rationality

Stefan Schreieder

Vol. 12 (2018), No. 2, 479–490
Abstract

We prove a general specialization theorem which implies stable irrationality for a wide class of quadric surface bundles over rational surfaces. As an application, we solve, with the exception of two cases, the stable rationality problem for any very general complex projective quadric surface bundle over 2, given by a symmetric matrix of homogeneous polynomials. Both exceptions degenerate over a plane sextic curve, and the corresponding double cover is a K3 surface.

Keywords
rationality problem, stable rationality, decomposition of the diagonal, unramified cohomology, Brauer group, Lüroth problem
Mathematical Subject Classification 2010
Primary: 14E08, 14M20
Secondary: 14J35, 14D06
Milestones
Received: 24 June 2017
Revised: 8 November 2017
Accepted: 18 December 2017
Published: 13 May 2018
Authors
Stefan Schreieder
Mathematisches Institut
Ludwig-Maximilians-Universität München
München
Germany