Abstract
We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with the given field of rational functions and an infinitely transitive regular action of a group of algebraic automorphisms generated by unipotent algebraic subgroups. We expect that this property holds for all unirational varieties and in fact is a peculiar one for this class of algebraic varieties among those varieties which are rationally connected.
Mathematics Subject Classification codes (2000): 14M20, 14M17, 14R20
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Notes
- 1.
In the terminology of [1], if ∂ is a LND, \(g =\mathrm{ exp}(t\partial )\vert _{t=1}\) the corresponding automorphism, and f a function from the kernel of ∂, then the automorphism \(g_{1} =\mathrm{ exp}(tf\partial )\vert _{t=1}\) is called a replica of g.
- 2.
This question was suggested by J.-L. Colliot-Thélène in connection with Conjecture 1.4. However, there are reasons to doubt the positive answer, since, for example, it would imply that X is (stably) birationally isomorphic to G ∕ H, where both G, H are (finite dimensional) reductive algebraic groups. Even more, up to stable birational equivalence, we may assume that \(X = G^\prime/H^\prime\), where H′ is a finite group and G′ is the product of a general linear group, spinor groups, and exceptional Lie groups. The latter implies, among other things, that there are only countably many stable birational equivalence classes of unirational varieties, but we could not develop a rigorous argument to bring this to contradiction.
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Acknowledgements
The first author was supported by NSF grant DMS-1001662 and by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023. The third author was supported by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023, RFBR grants 12-01-31342 and 12-01-33101, by the “EADS Foundation Chair in Mathematics”, Russian-French Poncelet Laboratory (UMI 2615 of CNRS MK-983.2013.1), and Dmitry Zimin fund “Dynasty.” The first author wants to thank Sh. Kaliman and M. Zaidenberg for useful discussions. The second author would like to thank Courant Institute for hospitality. The second author has also benefited from discussions with I. Cheltsov, Yu. Prokhorov, V. Shokurov, and K. Shramov. The third author would like to thank I. Arzhantsev for fruitful discussions. The authors are grateful to the referee for valuable comments. Finally, the authors thank the organizers of the summer school and the conference in Yekaterinburg (2011), where the work on the article originated.
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Bogomolov, F., Karzhemanov, I., Kuyumzhiyan, K. (2013). Unirationality and Existence of Infinitely Transitive Models. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds) Birational Geometry, Rational Curves, and Arithmetic. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6482-2_4
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DOI: https://doi.org/10.1007/978-1-4614-6482-2_4
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