Cohomological obstructions to equivariant unirationality
Yuri Tschinkel, Zhijia Zhang
TL;DR
The paper develops cohomological obstructions to equivariant unirationality for finite group actions on rational varieties, emphasizing del Pezzo surfaces and Kummer quartic double solids. It introduces Amitsur invariants $\mathrm{Am}^2(X,G)=\mathrm{Im}(\delta_2)$ and $\mathrm{Am}^3(X,G)=\mathrm{Im}(\delta_3)$ via the Leray spectral sequence for $G$-actions, relating them to Bogomolov multipliers and Condition (A). The main results classify actions with nonzero $\mathrm{Am}^3$ and show that obstructions are often controlled by the quaternion group $\mathrm{Q}_8$, yielding explicit non-$G$-unirational cases, including cubic and certain degree-2 del Pezzo surfaces, as well as Kummer quartic double solids. Overall, the work highlights subtle cohomological invariants that obstruct equivariant unirationality beyond the presence of rational points, and demonstrates concrete obstructions in low-degree rational varieties.
Abstract
We study cohomological obstructions to equivariant unirationality, with special regard to actions of finite groups on del Pezzo surfaces and Fano threefolds.