#### Vol. 12, No. 2, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors' Interests Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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 ${ℙ}^{2}$, 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