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Embedding non-projective Mori dream space

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Abstract

This paper is devoted to extend some Hu–Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, \(\mathbb {Q}\)-factorial algebraic varieties with finitely generated class group and Cox ring, here called weak Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied, showing that, on the one hand, those which are complete and admitting low Picard number are always projective, hence Mori dream spaces in the sense of Hu–Keel. On the other hand, an example of a wMDS that does not admit any neat embedded sharp completion (i.e. Picard number preserving) into a complete toric variety is given, on the contrary of what Hu and Keel exhibited for a MDS. Moreover, termination of the Mori minimal model program for every divisor and a classification of rational contractions for a complete wMDS are studied, obtaining analogous conclusions as for a MDS. Finally, we give a characterization of wMDS arising from a small \(\mathbb {Q}\)-factorial modification of a projective weak \(\mathbb {Q}\)-Fano variety.

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Notes

  1. In [30] we assumed \(\mathbb {C}\) as ground field. Actually all techniques and arguments therein employed are completely extendable to a general field \(\mathbb {K}=\overline{\mathbb {K}}\) with \({{\,\mathrm{char}\,}}\mathbb {K}=0\).

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Acknowledgements

First of all I would like to thank Lea Terracini, with whom several parts of the present work have been discussed. Her patience, criticism and encouragement have been an invaluable help in realizing this paper. Moreover, I would like to thank Cinzia Casagrande for pointing me out her precious notes [6] and for some useful remark. Finally, many thanks are due to Antonio Laface for several fruitful hints.

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  1. Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123, Turin, Italy

    Michele Rossi

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  1. Michele Rossi
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The author was partially supported by I.N.D.A.M. (Istituto Nazionale d’Alta Matematica) as a member of the G.N.S.A.G.A. (Gruppo Nazionale per le Strutture Algebriche, Geometriche e loro Applicazioni).

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Rossi, M. Embedding non-projective Mori dream space. Geom Dedicata 207, 355–393 (2020). https://doi.org/10.1007/s10711-019-00503-8

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