# Unirationality and Existence of Infinitely Transitive Models

## 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.

## Keywords

Open Subset Complete Intersection Algebraic Variety Toric Variety Ample Line Bundle## Notes

### 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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