# On the rationality problem for quadric bundles

## TL;DR

This paper classifies when complex quadric bundles over rational varieties are (stably) non-rational, extending the specialization method to prove non-rationality without universally CH_0-trivial resolutions.

## Contribution

It introduces a generalized specialization method that simplifies proving non-rationality of quadric bundles over rational varieties.

## Key findings

- Certain quadric bundles are not stably rational for specific dimensions and ranks.
- A new technique avoids the need for universally CH_0-trivial resolutions.
- The classification covers all positive integers n and r for these bundles.

## Abstract

We classify all positive integers n and r such that (stably) non-rational complex r-fold quadric bundles over rational n-folds exist. We show in particular that for any n and r, a wide class of smooth r-fold quadric bundles over projective n-space are not stably rational if r lies in the interval from $2^{n-1}-1$ to $2^{n}-2$. In our proofs we introduce a generalization of the specialization method of Voisin and Colliot-Th\'el\`ene--Pirutka which avoids universally $CH_0$-trivial resolutions of singularities.