Nikita A. Karpenko: Incompressibility of generic torsors of norm tori

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Submission: 2012, Aug 8

Let p be a prime integer, F a field of characteristic not p, T the norm torus of a degree pn extension field of F, and E a T-torsor over F such that the degree of each closed point on E is divisible by pn (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.

2010 Mathematics Subject Classification: 14L17; 14C25

Keywords and Phrases: Algebraic tori, toric varieties, incompressibility, Chow groups and Steenrod operations.

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