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Rationality problems and conjectures of Milnor and Bloch–Kato

Part of: (Co)homology theory Special varieties Forms and linear algebraic groups Higher algebraic $K$-theory Homological methods (field theory)

Published online by Cambridge University Press:  03 June 2013

Aravind Asok
Aravind Asok*
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA email [email protected]
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Abstract

We show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number $\ell $ and any integer $n\geq 2$, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree $n$ unramified étale cohomology class with $\ell $-torsion coefficients. When $\ell = 2$, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified étale cohomology class of lower degree.

Keywords

rationalityunramified cohomologyMilnor conjecture𝔸1-connectedness

MSC classification

Secondary: 14M20: Rational and unirational varieties 19D45: Higher symbols, Milnor $K$-theory 11E81: Algebraic theory of quadratic forms; Witt groups and rings 12G05: Galois cohomology 14F42: Motivic cohomology; motivic homotopy theory
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 8 , August 2013 , pp. 1312 - 1326
Copyright
© The Author(s) 2013 

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