Retract (-$mathit{i}$) rationality and its necessary conditions expressed by unramified presheaves : Noether's problem of a finite group $G$ as an example (Algebraic Number Theory and Related Topics)

Abstract

This is another short introduction to the author's study of the rationality problem, which centers the hierarchies of the form: lower rationality = higher ruledness. A particular emphasis is given for Noether's problem of a finite group G, where a technical difficult emerges because the relevant geometric object BG is not approximated by smooth proper varieties. The author's novelty here is a construction of the stable birational subsheaf for any unramified presheaf in the sense of Morel. This gives us a very strong necessary condition for retract (-i) rationality of smooth, not necessary proper, varieties over a perfect field.