Incompressibility of generic torsors of norm tori

N. A. Karpenko

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 25, Tome 414 (2013), pp. 106-112

Résumé

Let $p$ be a prime integer, $F$ be a field of characteristic not $p$, $T$ the norm torus of a degree $p^n$ extension field of $F$, and $E$ a $T$-torsor over $F$ such that the degree of each closed point on $E$ is divisible by $p^n$ (a generic $T$-torsor has this property). We prove that $E$ is $p$-incompressible. Moreover, all smooth compactifications of $E$ (including those given by toric varieties) are $p$-incompressible. The main requisites of the proof are: (1) A. Merkurjev's degree formula (requiring the characteristic assumption), generalizing M. Rost's degree formula, and (2) combinatorial construction of a smooth projective fan invariant under an action of a finite group on the ambient lattice due to J.-L. Colliot-Thélène–D. Harari–A. N. Skorobogatov, produced by refinement of J.-L. Brylinski's method with a help of an idea of K. Künnemann.