Cohomologie non ramifi\'ee de degr\'e 3 : vari\'et\'es cellulaires et surfaces de del Pezzo de degr\'e au moins 5

TL;DR

This paper investigates the third unramified cohomology of certain algebraic varieties over fields of characteristic zero, showing finiteness or triviality of specific quotients for varieties related to rational surfaces and del Pezzo surfaces of degree at least 5.

Contribution

It proves the finiteness of the quotient of the third unramified cohomology for universal torsors over rational surfaces and the triviality of this quotient for most del Pezzo surfaces of degree ≥ 5.

Findings

The quotient is finite for universal torsors over rational surfaces.

The quotient is zero for del Pezzo surfaces of degree ≥ 5, except for a special case.

Provides new insights into unramified cohomology of cellular varieties.

Abstract

We consider geometrically cellular varieties $X$ over an arbitrary field of characteristic zero. We study the quotient of the third unramified cohomology group $H^3_{nr}(X,\mathbb{Q}/\mathbb{Z}(2))$ by its constant part. For $X$ a smooth compactification of a universal torsor over a geometrically rational surface, we show that this quotient if finite. For $X$ a del Pezzo surface of degree $\geq 5$, we show that this quotient is zero, unless $X$ is a del Pezzo surface of degree 8 of a special type.