An Artin--Mumford criterion for conic bundles in characteristic two

TL;DR

The paper extends the Artin--Mumford irrationality criterion to characteristic $2$ by producing a $2$-torsion class in crystalline cohomology from a conic bundle with a disconnected discriminant, under hypotheses including $H^2(B,\Omega^1_{B/k})=0$ and the presence of nonreduced fibers on each discriminant component. By leveraging crystalline cohomology and degeneration techniques à la Voisin, it derives irrationality results in characteristic $0$ and provides explicit examples of irrational conic bundles in characteristic two. The authors develop a cohomological framework for conic bundles, compute the relevant differential-form cohomology, and show how discriminant data yields a torsion obstruction to the decomposition of the diagonal. They also introduce elementary transformations to separate discriminant components and present concrete AM-type examples, including very general families over $\mathbb P^2$, to obtain stable irrationality results with potential extensions to higher dimensions.

Abstract

We prove a characteristic two version of the famous criterion of Artin and Mumford for irrationality of conic bundles. On the one hand, combined with the pathological behaviour of conic bundles in characteristic two, this allows us to construct easier and more explicit examples of irrational conic bundles. On the other hand, degeneration techniques à la Voisin allow to deduce irrationality results in characteristic zero.

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Definition 3.4
• Remark 3.5