Skip to main content Accessibility help
×
×
Home

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

  • RYO YAMAGISHI (a1)

Abstract

We prove that a quotient singularity ℂn/G by a finite subgroup GSLn(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.

Copyright

References

Hide All
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).
2. Arzhantsev, I. V. and Gaĭfullin, S. A., Cox rings, semigroups, and automorphisms of affine varieties, Math. Sb. 201 (1) (2010), 324; translation in Sb. Math. 201(1–2) (2010), 1–21.
3. Andreatta, M. and Wiśniewski, J. A., 4-dimensional symplectic contractions, Geom. Dedicata 168 (2014), 311337.
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), 405468.
5. Bellamy, G. On singular Calogero–Moser spaces, Bull. Lond. Math. Soc. 41 (2) (2009), 315326.
6. Benson, D. J., Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, vol. 190 (Cambridge University Press, Cambridge, UK, 1993).
7. Berchtold, F. and Hausen, J., GIT equivalence beyond the ample cone, Michigan Math. J. 54 (3) (2006), 483515.
8. Berchtold, F. and Hausen, J., Cox rings and combinatorics, Trans. Amer. Math. Soc. 359 (3) (2007), 12051252.
9. Bridgeland, T., King, A. and Reid, M., The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (3) (2001), 535554.
10. Bellamy, G. and Schedler, T., A new linear quotient of C4 admitting a symplectic resolution, Math. Zeit. 273 (3–4) (2013), 753769.
11. Bellamy, G. and Schedler, T., On the (non)existence of symplectic resolutions of linear quotients, Math. Res. Lett. 23 (6) (2016), 15371564.
12. Cohen, A. M., Finite quaternionic reflection groups, J. Algebra 64 (2) (1980), 293324.
13. Donten-Bury, M., Cox rings of minimal resolutions of surface quotient singularities, Glasg. Math. J. 58 (2) (2016), 325355.
14. Donten-Bury, M. and Grab, M., Cox rings of some symplectic resolutions of quotient singularities, arXiv:1504.07463v2.
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), 395434.
16. Facchini, L., González-Alonso, V. and Lason, M., Cox rings of Du Val singularities, Le Mat. 66 (2) (2011), 115136.
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.de
18. Grayson, D. and Stillman, M., Macaulay 2: A software system for research in algebraic geometry; available at http://www.math.uiuc.edu/Macaulay2.
19. Hu, Y. and Keel, S., Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331348.
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), 221240.
21. Kaledin, D., Multiplicative McKay correspondence in the symplectic case, arXiv:0311409v2.
22. Kaledin, D., On crepant resolutions of symplectic quotient singularities. Sel. Math. (N.S.), 9 (4) (2003), 529555.
23. Kollár, J., Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1) (1993), 177215.
24. Lehn, M. and Sorger, C., A symplectic resolution for the binary tetrahedral group, Séminaires Congres 25 (2010) 427433.
25. Milne, J. S., Lectures on Étale Cohomology, version 2.21, available at http://www.jmilne.org/math/CourseNotes/lec.html.
26. Mumford, D., Fogarty, J. and Kirwan, F.. Geometric invariant theory, 3rd edition (Springer Verlag, New York, 1994).
27. Roan, S.-S., Minimal resolutions of Gorenstein orbifolds in dimension three, Topology 35 (2) (1996), 489508.
28. Shephard, G. C. and Todd, J. A., Finte unitary reflection groups, Canad. J. Math. 6 (1954), 274304.
29. Thaddeus, M., Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (3) (1996), 691723.
30. Verbitsky, M., Holomorphic symplectic geometry and orbifold singularities, Asian J. Math. 4 (3) (2000), 553563.
31. Watanabe, K., Certain invariant subrings are Gorenstein. I, II, Osaka J. Math. 11 (1974), 18; ibid. 11 (1974), 379–388.
32. Wierzba, J. and Wiśniewski, J. A., Small contractions of symplectic 4-folds, Duke Math. J. 120 (1) (2003), 6595.
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), 465488.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed