Incompressibility of Generic Torsors of Norm Tori

Let p be a prime integer, F a field of characteristic not p, T the norm torus of a degree p 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 (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. ColliotThélène D. Harari A.N. Skorobogatov, produced by refinement of J.-L. Brylinski’s method with a help of an idea of K. Künnemann. Let F be a field, p a prime integer. We say that an F -variety (by which we mean just a separated F -scheme of finite type) is p-incompressible (resp., incompressible), if its canonical p-dimension (resp., canonical dimension), defined as in [13, §4b], is equal to its usual dimension. An integral variety X is incompressible if and only if any rational map X 99K X is dominant, [13, Proposition 4.3]; p-incompressibility is a p-local version of incompressibility implying the incompressibility. Given an arbitrary F -variety V , we write nV for the greatest common divisor of the degrees of the closed points on V . Usually, we are only interested in vp(nV ), where vp is the p-adic valuation. By a compactification of an F -variety V we mean a complete F -variety X containing a dense open subvariety isomorphic to V . Given a finite separable extension field (or, more generally, an étale algebra) K/F , its norm torus T = TK/F , also called norm one torus and usually denoted by R K/F (Gm), is the algebraic torus defined as the kernel of the norm map of algebraic tori NK/F : RK/F (Gm) → Gm, F , where RK/F is the Weil transfer with respect to K/F . The group of F -points of T is the subgroup of norm 1 elements in K×. As a variety, T is the hypersurface in the affine F -space K given by the equation NK/F = 1. Date: 5 August 2012. Revised: 25 January 2013.