Weakly-exceptional quotient singularities

Dmitrijs Sakovics

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 885-902
  • ISSN: 2391-5455

Abstract

top
A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow-up. In dimension 2, V. Shokurov proved that weakly-exceptional quotient singularities are exactly those of types D n, E 6, E 7, E 8. This paper classifies the weakly-exceptional quotient singularities in dimensions 3 and 4.

How to cite

top

Dmitrijs Sakovics. "Weakly-exceptional quotient singularities." Open Mathematics 10.3 (2012): 885-902. <http://eudml.org/doc/269805>.

@article{DmitrijsSakovics2012,
abstract = {A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow-up. In dimension 2, V. Shokurov proved that weakly-exceptional quotient singularities are exactly those of types D n, E 6, E 7, E 8. This paper classifies the weakly-exceptional quotient singularities in dimensions 3 and 4.},
author = {Dmitrijs Sakovics},
journal = {Open Mathematics},
keywords = {Quotient singularities; Weakly-exceptional singularities; Log canonical threshold; exceptional singularities; weakly exceptional singularities; log canonical; log terminal; plt; quotient singularities},
language = {eng},
number = {3},
pages = {885-902},
title = {Weakly-exceptional quotient singularities},
url = {http://eudml.org/doc/269805},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Dmitrijs Sakovics
TI - Weakly-exceptional quotient singularities
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 885
EP - 902
AB - A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow-up. In dimension 2, V. Shokurov proved that weakly-exceptional quotient singularities are exactly those of types D n, E 6, E 7, E 8. This paper classifies the weakly-exceptional quotient singularities in dimensions 3 and 4.
LA - eng
KW - Quotient singularities; Weakly-exceptional singularities; Log canonical threshold; exceptional singularities; weakly exceptional singularities; log canonical; log terminal; plt; quotient singularities
UR - http://eudml.org/doc/269805
ER -

References

top
  1. [1] Birkar C., Cascini P., Hacon C.D., McKernan J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 2010, 23(2), 405–468 http://dx.doi.org/10.1090/S0894-0347-09-00649-3 Zbl1210.14019
  2. [2] Blichfeldt H., Finite Collineation Groups, The University of Chicago Science Series, University of Chicago Press, Chicago, 1917 
  3. [3] Bruce J.W., Wall C.T.C., On the classification of cubic surfaces. J. London Math. Soc., 1979, 19(2), 245–256 http://dx.doi.org/10.1112/jlms/s2-19.2.245 Zbl0393.14007
  4. [4] Cheltsov I., Log canonical thresholds of del Pezzo surfaces, Geom. Funct. Anal., 2008, 18(4), 1118–1144 http://dx.doi.org/10.1007/s00039-008-0687-2 Zbl1161.14030
  5. [5] Chel’tsov I., Shramov C., Log canonical thresholds of smooth Fano threefolds, Russian Math. Surveys, 2008, 63(5), 859–958 http://dx.doi.org/10.1070/RM2008v063n05ABEH004561 Zbl1167.14024
  6. [6] Cheltsov I., Shramov C., On exceptional quotient singularities, Geom. Topol., 2011, 15(4), 1843–1882 http://dx.doi.org/10.2140/gt.2011.15.1843 Zbl1232.14001
  7. [7] Cheltsov I., Shramov C., Six-dimensional exceptional quotient singularities, preprint avaialble at http://arxiv.org/pdf/1001.3863.pdf Zbl1281.14004
  8. [8] Demailly J.-P., On Tian’s invariant and log canonical thresholds, Appendix to: Chel’tsov I., Shramov C., Log canonical thresholds of smooth Fano threefolds, Russian Math. Surveys, 2008, 63(5), 859–958 http://dx.doi.org/10.1070/RM2008v063n05ABEH004561 
  9. [9] Dolgachev I.V., Iskovskikh V.A., Finite subgroups of the plane Cremona group, In: Algebra, Arithmetic, and Geometry. I, Progr. Math., 269, Birkhäuser, Boston, 443–548 Zbl1219.14015
  10. [10] Flannery D.L., The finite irreducible monomial linear groups of degree 4, J. Algebra, 1999, 218(2), 436–469 http://dx.doi.org/10.1006/jabr.1999.7883 Zbl0939.20050
  11. [11] Frobenius F.G., Über den von L. Bieberbach gefundenen Beweis eines Satzes von C. Jordan, Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1911, 241–248 Zbl42.0152.01
  12. [12] Höfling B., Finite irreducible imprimitive nonmonomial complex linear groups of degree 4, J. Algebra, 2001, 236(2), 419–470 http://dx.doi.org/10.1006/jabr.2000.8525 Zbl1079.20501
  13. [13] Hosoh T., Automorphism groups of cubic surfaces, J. Algebra, 1997, 192(2), 651–677 http://dx.doi.org/10.1006/jabr.1996.6968 
  14. [14] Iskovskikh V.A., Prokhorov Y.G., Fano Varieties, In: Algebraic Geometry, 5, Encyclopaedia Math. Sci., 47, Springer, Berlin, 1999 
  15. [15] Kollár J., Singularities of pairs, In: Algebraic Geometry, Santa Cruz, July 9–29, 1995, Proc. Sympos. Pure Math., 62(1), American Mathematical Society, Providence, 1997, 221–287 Zbl0905.14002
  16. [16] Kudryavtsev S.A., Pure log terminal blow-ups, Math. Notes, 2001, 69(5–6), 814–819 http://dx.doi.org/10.1023/A:1010234532502 Zbl1015.14007
  17. [17] Markushevich D., Prokhorov Yu.G., Exceptional quotient singularities, Amer. J. Math., 1999, 121(6), 1179–1189 http://dx.doi.org/10.1353/ajm.1999.0044 Zbl0958.14003
  18. [18] Miller G.A., Blichfeldt H.F., Dickson L.E., Theory and Applications of Finite Groups, Dover, New York, 1961 Zbl0098.25103
  19. [19] Prokhorov Yu.G., Blow-ups of canonical singularities, In: Proceedings of the International Algebraic Conference held on the occasion of the 90th birthday of A.G.Kurosh, Moscow, May 25–30, 1998, Walter de Gruyter, Berlin, 2000, 301–317 
  20. [20] Prokhorov Yu.G., Sparseness of exceptional quotient singularities, Math. Notes, 2000, 68(5–6), 664–667 http://dx.doi.org/10.1023/A:1026636027672 Zbl1060.14521
  21. [21] Segre B., The Non-Singular Cubic Surfaces, Oxford University Press, Oxford, 1942 Zbl68.0358.01
  22. [22] Shokurov V.V., 3-fold log flips, Russian Acad. Sci. Izv. Math., 1993, 40(1), 95–202 http://dx.doi.org/10.1070/IM1993v040n01ABEH001862 Zbl0785.14023
  23. [23] Springer T.A., Invariant Theory, Lecture Notes in Math., 585, Springer, Berlin-New York, 1977 
  24. [24] Tian G., On Kähler-Einstein metrics on certain Kähler manifolds with C 1(M) > 0, Invent. Math., 1987, 89(2), 225–246 http://dx.doi.org/10.1007/BF01389077 Zbl0599.53046
  25. [25] Tian G., Yau S.-T., Kähler-Einstein metrics on complex surfaces with C 1 > 0, Comm. Math. Phys., 1987, 112(1), 175–203 http://dx.doi.org/10.1007/BF01217685 Zbl0631.53052
  26. [26] Yau S.S.-T., Yu Y., Gorenstein Quotient Singularities in Dimension Three, Mem. Amer. Math. Soc., 505, American Mathematical Society, Providence, 1993 Zbl0799.14001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.