# Quadric surface bundles over surfaces and stable rationality

## TL;DR

This paper proves a general theorem showing many quadric surface bundles over rational surfaces are stably irrational, and applies it to classify stable rationality for bundles over the projective plane, with two special degenerate cases involving K3 surfaces.

## Contribution

It introduces a new specialization theorem that determines stable irrationality of quadric surface bundles over rational surfaces, advancing the understanding of their rationality properties.

## Key findings

- Most quadric surface bundles over rational surfaces are stably irrational.
- The stable rationality problem is solved for most bundles over the projective plane.
- Two special cases degenerate to K3 surfaces, remaining unresolved.

## 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 the projective plane, 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.