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A software package for Mori dream spaces

Part of: Computational aspects in algebraic geometry

Published online by Cambridge University Press:  01 November 2015

Jürgen Hausen  and
Simon Keicher
Jürgen Hausen
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany email juergen.hausen@uni-tuebingen.de
Simon Keicher
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany email keicher@mail.mathematik.uni-tuebingen.de

Abstract

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Mori dream spaces form a large example class of algebraic varieties, comprising the well-known toric varieties. We provide a first software package for the explicit treatment of Mori dream spaces and demonstrate its use by presenting basic sample computations. The software package is accompanied by a Cox ring database which delivers defining data for Cox rings and Mori dream spaces in a suitable format. As an application of the package, we determine the common Cox ring for the symplectic resolutions of a certain quotient singularity investigated by Bellamy–Schedler and Donten-Bury–Wiśniewski.

MSC classification

Primary: 14Q10: Surfaces, hypersurfaces 14Q15: Higher-dimensional varieties
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 647 - 659
Copyright
© The Author(s) 2015 

References

Artebani, M. and Laface, A., ‘Hypersurfaces in Mori dream spaces’, J. Algebra 371 (2012) 26–37.CrossRefGoogle Scholar
Arzhantsev, I., Derenthal, U., Hausen, J. and Laface, A., Cox rings , Cambridge Studies in Advanced Mathematics 144 (Cambridge University Press, Cambridge, 2015).Google Scholar
Arzhantsev, I. and Gaĭfullin, S., ‘Cox rings, semigroups, and automorphisms of affine varieties’, Math. Sb. 201 (2010) no. 1, 3–24.Google Scholar
Bechtold, B., Hausen, J., Huggenberger, E. and Nicolussi, M., ‘On terminal Fano 3-folds with 2-torus action’, Int. Math. Res. Not. IMRN, doi:10.1093/imrn/rnv190.Google Scholar
Bellamy, G. and Schedler, T., ‘A new linear quotient of C 4 admitting a symplectic resolution’, Math. Z. 273 (2013) no. 3–4, 753–769.Google Scholar
Berchtold, F. and Hausen, J., ‘Cox rings and combinatorics’, Trans. Amer. Math. Soc. 359 (2007) no. 3, 1205–1252.Google Scholar
Berchtold, F., Hausen, J., Keicher, S., Vollmert, R. and Widmann, M., ‘TorDiv 1.4 — a Maple package on toric geometry and geometric invariant theory’, available at http://www.mathematik.uni-tuebingen.de/∼hausen/.Google Scholar
Birkar, C., Cascini, P., Hacon, C. and McKernan, J., ‘Existence of minimal models for varieties of log general type’, J. Amer. Math. Soc. 23 (2010) 405–468.Google Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) no. 3–4, 235–265.Google Scholar
Brown, G., ‘A database of polarized K3 surfaces’, Experiment. Math. 16 (2007) no. 1, 7–20.CrossRefGoogle Scholar
Derenthal, U., ‘Singular Del Pezzo surfaces whose universal torsors are hypersurfaces’, Proc. Lond. Math. Soc. (3) 108 (2014) no. 3, 638–681.Google Scholar
Donten-Bury, M. and Wiśniewski, J., ‘On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32’, Preprint, 2014, arXiv:1409.4204v3.Google Scholar
Franz, M., ‘Convex – a Maple package for convex geometry’, available at http://www.math.uwo.ca/∼mfranz/convex/ and the Maple homepage http://www.maplesoft.com/applications/view.aspx?SID=4507.Google Scholar
Gutsche, S., ‘GAP package ToricVarieties’, available at http://www.gap-system.org/Packages/toricvarieties.html.Google Scholar
Hausen, J., ‘Cox rings and combinatorics II’, Mosc. Math. J. 8 (2008) no. 4, 711–757, 847.Google Scholar
Hausen, J. and Keicher, S., ‘Cox ring database’, available at http://www.math.uni-tuebingen.de/user/keicher/coxringdb/.Google Scholar
Hausen, J. and Keicher, S., ‘MDSpackage – a package for Mori dream spaces’, available at http://www.math.uni-tuebingen.de/user/keicher/MDS/.Google Scholar
Hausen, J., Keicher, S. and Laface, A., ‘Computing Cox rings’, Math. Comp. 85 (2016) no. 297, 467–502.Google Scholar
Hu, Y. and Keel, S., ‘Mori dream spaces and GIT’, Michigan Math. J. 48 (2000) 331–348; dedicated to William Fulton on the occasion of his 60th birthday.Google Scholar
Iano-Fletcher, A. R., ‘Working with weighted complete intersections’, Explicit birational geometry of 3-folds , London Mathematical Society Lecture Note Series 281 (Cambridge University Press, 2000) 101–173.Google Scholar
Jow, S.-Y., ‘A Lefschetz hyperplane theorem for Mori dream spaces’, Math. Z. 268 (2011) no. 1–2, 197–209.CrossRefGoogle Scholar
Joyner, D., ‘GAP package toric’, available at http://www.gap-system.org/Packages/toric.html.Google Scholar
Kastner, L., Lorenz, B., Paffenholz, A. and Winz, A.-L., polymake_toric, available at https://github.com/lkastner/polymake_toric.Google Scholar
Keicher, S., ‘Computing the GIT-fan’, Int. J. Algebra Comput. 22 (2012) no. 7, 1250064, 11 pp.Google Scholar
Keicher, S., ‘Algorithms for Mori dream spaces’, PhD Thesis, Universität Tübingen, 2014, http://nbn-resolving.de/urn:nbn:de:bsz:21-dspace-540614.Google Scholar
Maple 10, Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario. See http://www.maplesoft.com/.Google Scholar
McKernan, J., ‘Mori dream spaces’, Jpn. J. Math. 5 (2010) no. 1, 127–151.CrossRefGoogle Scholar
Verrill, H. and Joyner, D., ‘Computing with toric varieties’, J. Symbolic Comput. 42 (2007) no. 5, 511–532.CrossRefGoogle Scholar