Degree 2 del Pezzo surface bundles and stable rationality

TL;DR

This paper investigates the arithmetic properties of degree 2 del Pezzo surface bundles over rational surfaces, revealing instances where stable rationality fails due to nontrivial unramified cohomology.

Contribution

It introduces a method to analyze the Picard group homomorphism and constructs examples of del Pezzo surface fibrations with nontrivial unramified cohomology, demonstrating failure of stable rationality.

Findings

Constructed examples with nontrivial unramified cohomology

Demonstrated failure of stable rationality in certain degenerations

Linked Picard group cokernel to rationality properties

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 $\pi:X\to S$ in del Pezzo surfaces of degree 2 over a rational surface $S$, we construct examples with nontrivial relative unramified cohomology group $H^2_{nr,\pi}(k(X)/k)$. A specialization argument implies the failure of stable rationality of varieties specializing to $X$.