Degree 2 del Pezzo surface bundles and stable rationality

TL;DR

The work develops a framework for proving non-stable rationality of fibrations by del Pezzo surfaces of degree $2$ over rational bases via relative unramified cohomology, constructing a reference variety with a nontrivial relative class $H^2_{nr,\pi}(k(X)/k)$ to apply a refined specialization argument. It exhibits a concrete reference fibration and derives a corollary: very general double covers of $\mathbb{P}^2\times\mathbb{P}^2$ branched along a divisor of bidegree $(2q,4)$ are not stably rational for $q\ge 4$, over an uncountable algebraically closed field of characteristic not $2$. The paper also analyzes diagonal del Pezzo surfaces to compute the $G$-invariants of the Picard group and the behavior of the Brauer group morphisms, establishing precise kernels or injectivity that feed into the non-stable rationality results.

Abstract

We study the arithmetic of del Pezzo surfaces $Y$ of degree 2 over a function field, and in particular, the cokernel of the homomorphism from the Picard group to the Galois-invariants of the geometric Picard group $\operatorname{Pic} Y \rightarrow(\operatorname{Pic} \bar{Y})^{G}$. Applying this to a fibration $π:X\to S$ in del Pezzo surfaces of degree 2 over a rational surface $S$, we construct examples with nontrivial relative unramified cohomology group $H^2_{nr,π}(k(X)/k)$. A specialization argument implies the failure of stable rationality of varieties specializing to $X$.

Theorems & Definitions (14)

• Theorem 1
• Proposition 2
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Corollary 6
• proof