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A Remark on Rationally Connected Varieties and Mori Dream Spaces

Part of: Birational geometry

Published online by Cambridge University Press:  26 September 2018

Claudio Fontanari  and
Diletta Martinelli
Claudio Fontanari*
Affiliation:
Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123 Povo, Trento, Italy ([email protected])
Diletta Martinelli
Affiliation:
Diletta Martinelli, School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK ([email protected])
*
*Corresponding author.
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Abstract

In this short note, we show that a construction by Ottem provides an example of a rationally connected variety that is not birationally equivalent to a Mori dream space with terminal singularities.

Keywords

Mori dream spacerationally connected varietybirational rigidity

MSC classification

Primary: 14E30: Minimal model program (Mori theory, extremal rays)
Secondary: 14E07: Birational automorphisms, Cremona group and generalizations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 259 - 263
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Ahmadinezhad, H. and Zucconi, F., Mori dream spaces and birational rigidity of Fano 3-folds, Adv. Math. (N. Y.) 292 (2016), 410–445.Google Scholar
2.Birkar, C., Cascini, P., Hacon, C. and McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23(2) (2010), 405–468.Google Scholar
3.Cheltsov, I., Birationally rigid Fano varieties, Russ. Math. Surv. 60(5) (2005), 875.Google Scholar
4.Cheltsov, I., On singular cubic surfaces, Asian J. Math. 13 (2009), 191–214.Google Scholar
5.Corti, A. and Mella, M., Birational geometry of terminal quartic 3-folds, I, Amer. J. Math. 126(4) (2004), 739–761.Google Scholar
6.Graber, T., Harris, J. and Starr, J., Families of rationally connected varieties, J. Amer. Math. Soc. 16(1) (2003), 57–67.Google Scholar
7.Hu, Y. and Keel, S., Mori dream spaces and GIT, Michigan Math. J. 48(1) (2000), 331–348.Google Scholar
8.Kollár, J., Singularities of pairs, Proceedings of Symposia in Pure Mathematics, Volume 62, pp. 221–288 (American Mathematical Society, 1997).Google Scholar
9.Kollár, J. and Mori, S., Birational geometry of algebraic varieties (Cambridge University Press, 2008).Google Scholar
10.Kollár, J., Miyaoka, Y. and Mori, S., Rational connectedness and boundedness of Fano manifolds, J. Diff. Geom. 36 (1992), 765–779.Google Scholar
11.Krylov, I., Rationally connected non-Fano type varieties. Preprint, arXiv:1406.3752v3, 2015.Google Scholar
12.Ottem, C., Birational geometry of hypersurfaces in products of projective spaces, Math. Z. 280(1–2) (2015), 135–148.Google Scholar
13.Pukhlikov, A., Birationally rigid varieties (American Mathematical Society, 2013).Google Scholar
14.Pukhlikov, A., Birationally rigid Fano fibre spaces. II, Izv. Math. 79(4) (2015), 809.Google Scholar
15.Tian, G., On Kähler–Einstein metrics on certain Kähler manifolds with c 1(M) > 0, Inventiones Math. 89(2) (1987), 225–246.+0,+Inventiones+Math.+89(2)+(1987),+225–246.>Google Scholar
16.Zhang, Q., Rational connectedness of log Q-Fano varieties, J. Reine Angew. Math. 2006(590) (2006), 131–142.Google Scholar