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ON SMOOTHNESS OF MINIMAL MODELS OF QUOTIENT SINGULARITIES BY FINITE SUBGROUPS OF SLn(ℂ)

Published online by Cambridge University Press:  28 January 2018

RYO YAMAGISHI
RYO YAMAGISHI*
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwakecho, Kyoto, 606-8502, Japan e-mail: ryo-yama@math.kyoto-u.ac.jp
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Abstract

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We prove that a quotient singularity ℂn/G by a finite subgroup G ⊂ SLn(ℂ) has a crepant resolution only if G is generated by junior elements. This is a generalization of the result of Verbitsky (Asian J. Math.4(3) (2000), 553–563). We also give a procedure to compute the Cox ring of a minimal model of a given ℂn/G explicitly from information of G. As an application, we investigate the smoothness of minimal models of some quotient singularities. Together with work of Bellamy and Schedler, this completes the classification of symplectically imprimitive quotient singularities that admit projective symplectic resolutions.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 3 , September 2018 , pp. 603 - 634
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Arzhantsev, I., Derenthal, U., Hausen, J. and Laface, A., Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144 (Cambridge University Press, Cambridge, UK, 2015).Google Scholar
2. Arzhantsev, I. V. and Gaĭfullin, S. A., Cox rings, semigroups, and automorphisms of affine varieties, Math. Sb. 201 (1) (2010), 3–24; translation in Sb. Math. 201(1–2) (2010), 1–21.CrossRefGoogle Scholar
3. Andreatta, M. and Wiśniewski, J. A., 4-dimensional symplectic contractions, Geom. Dedicata 168 (2014), 311–337.CrossRefGoogle Scholar
4. 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.CrossRefGoogle Scholar
5. Bellamy, G. On singular Calogero–Moser spaces, Bull. Lond. Math. Soc. 41 (2) (2009), 315–326.CrossRefGoogle Scholar
6. Benson, D. J., Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, vol. 190 (Cambridge University Press, Cambridge, UK, 1993).CrossRefGoogle Scholar
7. Berchtold, F. and Hausen, J., GIT equivalence beyond the ample cone, Michigan Math. J. 54 (3) (2006), 483–515.CrossRefGoogle Scholar
8. Berchtold, F. and Hausen, J., Cox rings and combinatorics, Trans. Amer. Math. Soc. 359 (3) (2007), 1205–1252.CrossRefGoogle Scholar
9. Bridgeland, T., King, A. and Reid, M., The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (3) (2001), 535–554.CrossRefGoogle Scholar
10. Bellamy, G. and Schedler, T., A new linear quotient of C4 admitting a symplectic resolution, Math. Zeit. 273 (3–4) (2013), 753–769.CrossRefGoogle Scholar
11. Bellamy, G. and Schedler, T., On the (non)existence of symplectic resolutions of linear quotients, Math. Res. Lett. 23 (6) (2016), 1537–1564.CrossRefGoogle Scholar
12. Cohen, A. M., Finite quaternionic reflection groups, J. Algebra 64 (2) (1980), 293–324.CrossRefGoogle Scholar
13. Donten-Bury, M., Cox rings of minimal resolutions of surface quotient singularities, Glasg. Math. J. 58 (2) (2016), 325–355.CrossRefGoogle Scholar
14. Donten-Bury, M. and Grab, M., Cox rings of some symplectic resolutions of quotient singularities, arXiv:1504.07463v2.Google Scholar
15. Donten-Bury, M. and Wiśniewski, J. A., On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32, Kyoto J. Math. 57 (2) (2017), 395–434.CrossRefGoogle Scholar
16. Facchini, L., González-Alonso, V. and Lason, M., Cox rings of Du Val singularities, Le Mat. 66 (2) (2011), 115–136.Google Scholar
17. Greuel, G.-M., Pfister, G. and Schönemann, H.. Singular 3-1-6. A computer algebra system for polynomial computations. (Centre for Computer Algebra, University of Kaiserslautern, 2001). http://www.singular.uni-kl.deGoogle Scholar
18. Grayson, D. and Stillman, M., Macaulay 2: A software system for research in algebraic geometry; available at http://www.math.uiuc.edu/Macaulay2.Google Scholar
19. Hu, Y. and Keel, S., Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348.CrossRefGoogle Scholar
20. Ito, Y. and Reid, M., The McKay correspondence for finite subgroups of SL(3, C), in Higher-dimensional complex varieties (Trento, 1994) (de Gruyter, Berlin, 1996), 221–240.Google Scholar
21. Kaledin, D., Multiplicative McKay correspondence in the symplectic case, arXiv:0311409v2.Google Scholar
22. Kaledin, D., On crepant resolutions of symplectic quotient singularities. Sel. Math. (N.S.), 9 (4) (2003), 529–555.CrossRefGoogle Scholar
23. Kollár, J., Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1) (1993), 177–215.CrossRefGoogle Scholar
24. Lehn, M. and Sorger, C., A symplectic resolution for the binary tetrahedral group, Séminaires Congres 25 (2010) 427–433.Google Scholar
25. Milne, J. S., Lectures on Étale Cohomology, version 2.21, available at http://www.jmilne.org/math/CourseNotes/lec.html.Google Scholar
26. Mumford, D., Fogarty, J. and Kirwan, F.. Geometric invariant theory, 3rd edition (Springer Verlag, New York, 1994).CrossRefGoogle Scholar
27. Roan, S.-S., Minimal resolutions of Gorenstein orbifolds in dimension three, Topology 35 (2) (1996), 489–508.CrossRefGoogle Scholar
28. Shephard, G. C. and Todd, J. A., Finte unitary reflection groups, Canad. J. Math. 6 (1954), 274–304.Google Scholar
29. Thaddeus, M., Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (3) (1996), 691–723.CrossRefGoogle Scholar
30. Verbitsky, M., Holomorphic symplectic geometry and orbifold singularities, Asian J. Math. 4 (3) (2000), 553–563.CrossRefGoogle Scholar
31. Watanabe, K., Certain invariant subrings are Gorenstein. I, II, Osaka J. Math. 11 (1974), 1–8; ibid. 11 (1974), 379–388.Google Scholar
32. Wierzba, J. and Wiśniewski, J. A., Small contractions of symplectic 4-folds, Duke Math. J. 120 (1) (2003), 65–95.CrossRefGoogle Scholar
33. Yamagishi, R., Crepant resolutions of Slodowy slice in nilpotent orbit closure in $\mathfrak{sl}_N(\mathbb{C})$, Publ. Res. Inst. Math. Sci. 51 (3) (2015), 465–488.CrossRefGoogle Scholar