Computing automorphisms of Mori dream spaces

Hausen, Jürgen; Keicher, Simon; Wolf, Rüdiger

doi:10.1090/mcom/3185

Computing automorphisms of Mori dream spaces
HTML articles powered by AMS MathViewer

by Jürgen Hausen, Simon Keicher and Rüdiger Wolf PDF

Math. Comp. 86 (2017), 2955-2974 Request permission

Abstract:

We present an algorithm to compute the automorphism group of a Mori dream space. As an example calculation, we determine the automorphism groups of singular cubic surfaces with general parameters. The strategy is to study graded automorphisms of an affine algebra graded by a finitely generated abelian group and apply the results to the Cox ring. Besides the application to Mori dream spaces, our results could be used for symmetry based computing, e.g., for Gröbner bases or tropical varieties.

References

Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
Ivan Arzhantsev, Jürgen Hausen, Elaine Herppich, and Alvaro Liendo, The automorphism group of a variety with torus action of complexity one, Mosc. Math. J. 14 (2014), no. 3, 429–471, 641 (English, with English and Russian summaries). MR 3241755, DOI 10.17323/1609-4514-2014-14-3-429-471
Florian Berchtold and Jürgen Hausen, GIT equivalence beyond the ample cone, Michigan Math. J. 54 (2006), no. 3, 483–515. MR 2280492, DOI 10.1307/mmj/1163789912
J. N. S. Bidwell, M. J. Curran, and D. J. McCaughan, Automorphisms of direct products of finite groups, Arch. Math. (Basel) 86 (2006), no. 6, 481–489. MR 2241597, DOI 10.1007/s00013-005-1547-z
David A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17–50. MR 1299003
Ulrich Derenthal, Singular del Pezzo surfaces whose universal torsors are hypersurfaces, Proc. Lond. Math. Soc. (3) 108 (2014), no. 3, 638–681. MR 3180592, DOI 10.1112/plms/pdt041
Ulrich Derenthal, Jürgen Hausen, Armand Heim, Simon Keicher, and Antonio Laface, Cox rings of cubic surfaces and Fano threefolds, J. Algebra 436 (2015), 228–276. MR 3348473, DOI 10.1016/j.jalgebra.2015.04.028
Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012. A modern view. MR 2964027, DOI 10.1017/CBO9781139084437
Jean-Charles Faugère and Jules Svartz, Gröbner bases of ideals invariant under a commutative group: the non-modular case, ISSAC 2013—Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2013, pp. 347–354. MR 3206377, DOI 10.1145/2465506.2465944
Jürgen Hausen, Simon Keicher, and Antonio Laface, Computing Cox rings, Math. Comp. 85 (2016), no. 297, 467–502. MR 3404458, DOI 10.1090/mcom/2989
Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494, DOI 10.1307/mmj/1030132722
E. Huggenberger, Fano Varieties with Torus Action of Complexity One, PhD thesis, Universität Tübingen, Wilhelmstr. 32, 72074 Tübingen, 2013.
A. N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties, available at http://home.imf.au.dk/jensen/software/gfan/gfan.html.
Simon Keicher, Computing the GIT-fan, Internat. J. Algebra Comput. 22 (2012), no. 7, 1250064, 11. MR 2999370, DOI 10.1142/S0218196712500646
Yoshiyuki Sakamaki, Automorphism groups on normal singular cubic surfaces with no parameters, Trans. Amer. Math. Soc. 362 (2010), no. 5, 2641–2666. MR 2584614, DOI 10.1090/S0002-9947-09-05023-5
Stefan Steidel, Gröbner bases of symmetric ideals, J. Symbolic Comput. 54 (2013), 72–86. MR 3032638, DOI 10.1016/j.jsc.2013.01.005
Edward C. Turner and Daniel A. Voce, Tame automorphisms of finitely generated abelian groups, Proc. Edinburgh Math. Soc. (2) 41 (1998), no. 2, 277–287. MR 1626492, DOI 10.1017/S0013091500019647