Quadric surface bundles over surfaces and stable rationality

Schreieder, Stefan

doi:10.2140/ant.2018.12.479

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Quadric surface bundles over surfaces and stable rationality

Stefan Schreieder

Vol. 12 (2018), No. 2, 479–490

DOI: 10.2140/ant.2018.12.479

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

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

Vol. 12, No. 2, 2018