Unramified Brauer groups of conic bundle threefolds in characteristic two

We establish a formula for computing the unramified Brauer group of tame conic bundle threefolds in characteristic $2$. The formula depends on the arrangement and residue double covers of the discriminant components, the latter being governed by Artin-Schreier theory (instead of Kummer theory in characteristic not $2$). We use this to give new examples of threefold conic bundles defined over $\Bbb{Z}$ that are not stably rational over $\Bbb{C}$.