Advertisement

Results in Mathematics

, Volume 33, Issue 3–4, pp 208–265 | Cite as

On crepant resolutions of 2-parameter series of Gorenstein cyclic quotient singularities

  • Dimitrios I. Dais
  • Utz-Uwe Haus
  • Martin Henk
Research article

En

Abstract

Abstract

An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces ℂ r /G in dimensions r ≥ 4 would primarily demand the existence of projective, crepant, full desingularizations. Since this is not always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and sufficient conditions under which 2-parameter series of Gorenstein cyclic quotient singularities have torus-equivariant resolutions of this specific sort in all dimensions.

AMS subject classification

14B05 14M25 14Q15 32S05 52B20 

En]Keywords

cyclic quotient singularity Gorenstein singularity toric desingularization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Aguzzoli S., Mundici D.: An algorithmic desingularization of 3-dimensional toric varieties, Tôhoku Math. Jour. 46, (1994), 557–572.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Batyrev V.V.: Non-Archimedian integrals and stringy Euler numbers of log-terminal pairs, preprint, alg-geom / 9803071.Google Scholar
  3. [3]
    Batyrev V.V., Dais D.I.: Strong McKay correspondence, string theoretic Hodge numbers and mirror symmetry, Topology 35, (1996), 901–929.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Bouvier C, Gonzalez-Sprinberg G.: G-désingularisations de variétés toriques, C.R.Acad. sci. Paris 315, (1992), 817–820.MathSciNetMATHGoogle Scholar
  5. [5]
    Bouvier C., Gonzalez-Sprinberg G.: Systèm générateur, diviseurs essentiels et G-désingularisations de variétés toriques, Tôhoku Math. Jour. 47, (1995), 125–149.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Cohn H.: Support polygon and the resolution of modular functional singularities, Acta Arithmetica 24, (1973), 261–278.MathSciNetMATHGoogle Scholar
  7. [7]
    Dais D.I., Henk M.: On a series of Gorenstein cyclic quotient singularities admitting a unique projective crepant resolution, preprint, alg-geom / 9803094; to appear in: “Combinatorial Convex Geometry and Toric Varieties”, (ed. by G.Ewald & B.Teissier), Birkhäuser.Google Scholar
  8. [8]
    Dais D.I., Henk M., Ziegler G.M.: All abelian quotient c.i.-singularities admit projective crepant resolutions in all dimensions, preprint, alg-geom / 9704007.Google Scholar
  9. [9]
    Dais D.I., Henk M., Ziegler G.M.: On the existence of crepant resolutions of Gorenstein abelian quotient singularities in dimensions ≥ 4, in preparation.Google Scholar
  10. [10]
    Dixon J.D.: The number of steps in the Euclidean Algorithm, Jour. of Number Theory 2, (1970), 414–422.MATHCrossRefGoogle Scholar
  11. [11]
    Dixon L., Harvey J., Vafa C., Witten E.: Strings on orbifolds, I, II, Nuclear Phys. B, Vol. 261, (1985), 678–686, and Vol. 274, (1986), 285-314.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Du Val P.: On the singularities which do not affect the condition of adjunction I, II, III, Proc. Camb. Phil. Soc. 30, (1934), 453–459 & 483-491.CrossRefGoogle Scholar
  13. [13]
    Ewald G.: Combinatorial Convexity and Algebraic Geometry, Graduate Texts in Mathematics, Vol. 168, Springer-Verlag, (1996).Google Scholar
  14. [14]
    Finkel’shtein Y.Y.: Klein polygons and reduced continued fractions, Russian Math. Surveys 48, (1993), 198–200.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Firla R.T.: Hilbert-Cover- und Hulbert-Partitions-Probleme, Diplomarbeit, TU-Berlin, (1997).Google Scholar
  16. [16]
    Firla R.T., Ziegler G.M.: Hilbert bases, unimodular triangulations, and binary covers of rational polyhedral cones, preprint, (1997); to appear in Discrete & Comp. Geom.Google Scholar
  17. [17]
    Fujiki A.: On resolutions of cyclic quotient singularities, Publ. RIMS 10, (1974), 293–328.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Fulton W.: Introduction to Toric Varieties, Annals of Math. Studies, Vol. 131, Princeton University Press, (1993).Google Scholar
  19. [19]
    Gonzalez-Sprinberg G., Verdier J.L.: Construction géométrique de la correspondance de McKay, Ann. Sc. E.N.S. 16, (1983), 409–449.MathSciNetMATHGoogle Scholar
  20. [20]
    Gordan P.: Über die Auflösung linearer Gleichungen mit reellen Coefficienten, Math. Ann. 6, (1873), 23–28.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Haus U.-U.: A testing-program for the necessary and sufficient conditions of the theorem 5.15; see: http://www.zib.de/haus/2p-gor-cqs.html
  22. [22]
    Henk M., Weismantel R.: On Hilbert bases of polyhedral cones, preprint, (Konrad-Zuse-Zentrum, Berlin), SC 96-12, (1996); available from: http://www.zib-berlin.de/paperweb/paperweb?query=Henk&mode=all
  23. [23]
    Hibi T., Ohsugi H.: A normal (0,1)-polytope none of whose regular triangulations is unimodular, preprint, (1997); to appear in Discrete & Comp. Geom.Google Scholar
  24. [24]
    Hilbert D.: Über die Theorie der algebraischen Formen, Math. Ann. 36, (1890), 473–534. [See also: “Gesammelte Abhandlungen”, Band II, Springer-Verlag, (1933), 199-257.]MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    Hirzebruch F.: Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen, Math. Ann. 126, (1953), 1–22. [See also: “Gesammelte Abhandlungen”, Band I, Springer-Verlag, (1987), pp. 11-32.]MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Hirzebruch F.: Hilbert modular surfaces, Enseign. Math. 19, (1973), 183–281. [See also: “Gesammelte Abhandlungen”, Band II, Springer-Verlag, (1987), pp. 225-323.]MathSciNetMATHGoogle Scholar
  27. [27]
    Hirzebruch F., HöFER T.: On the Euler number of an orbifold, Math. Ann. 286, (1990), 255–260.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Ito Y., Nakamura I.: Hilbert schemes and simple singularities, preprint, (1997).Google Scholar
  29. [29]
    ITO Y., Reid M.: The McKay correspondence for finite subgroups of SL(3,C). In: “Higher Dimensional Complex Varieties”, Proceedings of the International Conference held in Trento, Italy, June 15-24, 1994, (edited by M.Andreatta, Th.Peternell); Walter de Gruyter, (1996), 221-240.Google Scholar
  30. [30]
    Jung H.W.E.: Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x, y in der Umbebung einer Stelle x = a, y = b, Jour, für die reine und ang. Math. 133, (1908), 289–314.MATHGoogle Scholar
  31. [31]
    Kempf G., Knudsen F., Mumford D., Saint-Donat D.: Toroidal Embeddings I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, (1973).Google Scholar
  32. [32]
    Kilian H.: Zur mittleren Anzahl von Schritten beim euklidischen Algorithmus, Elemente der Math. 38, (1983), 11–15.MathSciNetMATHGoogle Scholar
  33. [33]
    Klein F.: Vorlesungen über das Ikosaeder. Erste Ausg. Teubner Verlag, (1884). Zweite Ausg. von Teubner und Birkhäuser mit einer Einführung und mit Kommentaren von P. Slodowy, (1993). [English translation: Lectures on the icosahedron, Dover Pub. Co., (1956)].Google Scholar
  34. [34]
    Klein F.: Über die geometrische Auffassung der gewöhlichen Kettenbruchentwicklung, Nachrichten der Kgl. Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys. KL., Heft 3, (1895), 357–359. [See also: “Gesammelte Mathematische Abhandlungen”, Band II, herausg. von R.Fricke & H.Vermeil, Springer-Verlag, (1922), pp. 209-211.]Google Scholar
  35. [35]
    Klein F.: Ausgewählte Kapitel der Zahlentheorie, Bd. I. Vorlesungen 1895/96; herausgegeben von A.Sommerfeld, Göttingen, (1896) [see: Einleitung].Google Scholar
  36. [36]
    Klein F.: Elementarmathematik vom höheren Standpunkte aus, Grundlehren der Mathematischen Wissenschaften, Bd. 14, Dritte Auflage, Springer-Verlag, (1924); see pp. 46-48. [English translation: Elementary Mathematics from an advanced standpoint, Dover Pub. Co., (1932), pp. 42-44].Google Scholar
  37. [37]
    Knörrer H.: Group representations and the resolution of rational double points, Contemp. Math. 45, A.M.S., (1985), 175–222.CrossRefGoogle Scholar
  38. [38]
    Lagrange J.L.: Additions au mémoire sur la résolution des équation numériques, Mémoires de l’Acad. Roy. des Sciences et Belles-Lettres, t. 24, (1770); ∄VRES II, ed. by M.J.-A.Serret, Gauthier-Villars, Paris, (1868), pp. 581-652 [see i.p. §50-57, pp. 622-630].Google Scholar
  39. [39]
    Lamé G.: Note sur la limite du nobre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers, C.R. Acad. sci. Paris 19, (1844), 867–870.Google Scholar
  40. [40]
    Lamotke K.: Regular Solids and Isolated Singularities, Vieweg Adv. Lectures in Math., (1986).Google Scholar
  41. [41]
    McKay J.: Graphs, singularities and finite groups. In: “The Santa-Cruz Conference of Finite Groups”, Proc. of Symp. in Pure Math., A.M.S., Vol. 37, (1980), 183–186.MathSciNetCrossRefGoogle Scholar
  42. [42]
    Minkowski H.: Ueber die Annäherung an eine reelle Grösse durch rationale Zahlen, Math. Ann. 54, (1901), 91–124. [See also: “Gesammelte Abhandlungen”, Band I, herausg. von D.Hilbert, Teubner, (1911), pp. 320-356.]MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    Möbius A.F.: Beiträge zu der Lehre von den Kettenbrüchen, nebst einem Anhange dioptrischen Inhalts, Journal für die reine und ang. Math. 6, (1830), 215–243. [See also: “Gesammelte Werke”, Band II, herausg. von W.Scheibner & F.Klein, Verlag von S.Hirzel, Leipzig, (1887), pp. 505-539.]MATHCrossRefGoogle Scholar
  44. [44]
    Mohri K.: D-Branes and quotient singularities of Calabi-Yau fourfolds, preprint hep-th / 9707012.Google Scholar
  45. [45]
    Morrison D.R., Stevens G.: Terminal quotient singularities in dimension three and four, Proc. A.M.S. 90, (1984), 15–20.MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    Myerson G.: On semi-regular continued fractions, Archiv der Math. 48, (1987), 420–425.MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    Oda T.: Convex Bodies and Algebraic Geometry. An Introduction to the theory of toric varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 15, Springer-Verlag, (1988).Google Scholar
  48. [48]
    Oda T., Park H.S.: Linear Gale Transforms and GKZ-Decompositions, Tôhoku Math. Jour. 43, (1991), 375–399.MathSciNetMATHCrossRefGoogle Scholar
  49. [49]
    Perron O.: Die Lehre von den Kettenbrüchen, Bd. I, Dritte Auflage, Teubner, (1954).Google Scholar
  50. [50]
    Prill D.: Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. Jour. 34, (1967), 375–386.MathSciNetMATHCrossRefGoogle Scholar
  51. [51]
    Reid M.: Canonical threefolds, Journée de Géométrie Algébrique d’Angers, A. Beauville ed., Sijthoff and Noordhoff, Alphen aan den Rijn, (1980), 273-310.Google Scholar
  52. [52]
    Reid M.: Young person’s guide to canonical singularities. In: “Algebraic Geometry, Bowdoin 1985”, (edited by S.J.Bloch), Proc. of Symp. in Pure Math., A.M.S., Vol. 46, Part I, (1987), 345–416.CrossRefGoogle Scholar
  53. [53]
    Reid M.: The McKay correspondence and the physicist’s Euler number. Notes from lectures given at the Utah University (Sept. 1992) and at MSRI (Nov. 1992).Google Scholar
  54. [54]
    Reid M.: McKay correspondence, preprint, alg-geom / 9702016.Google Scholar
  55. [55]
    Riemenschneider O.: Deformationen von Quotientensingularitäten (nach zyklischen Gruppen), Math. Ann. 209, (1974), 211–248.MathSciNetMATHCrossRefGoogle Scholar
  56. [56]
    Sebö A.: Hilbert bases, Caratheodory’s theorem and combinatorial optimization, Proc. of the I.P.C.O. Conference, Waterloo, Canada, (1990), 431-455.Google Scholar
  57. [57]
    Shallit J.: Origins of the Analysis of the Euclidean Algorithm, Historia Math. 21, (1994), 401–419.MathSciNetMATHCrossRefGoogle Scholar
  58. [58]
    Stanley R.P.: Decompositions of rational convex polytopes, Annals of Discrete Math. 6, (1980), 333–342.MathSciNetCrossRefGoogle Scholar
  59. [59]
    Stanley R.P.: Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole Math. Series, (1986); second printing: Cambridge University Press, (1997).Google Scholar
  60. [60]
    Uspensky J.V., Heaslet M.A.: Elementary Number Theory, McGraw-Hill Book Co., Inc., (1939).Google Scholar
  61. [61]
    Vander Corput J.G.: Über Systeme von linear-homogenen Gleichungen und Ungleichungen, Konin-klijke Akademie van Wetenschappen te Amsterdam34, (1931), 368–371.Google Scholar
  62. [62]
    Van Der Corput J.G.: Über Diophantische Systeme von linear-homogenen Gleichungen und Ungleichungen, Koninklijke Akademie van Wetenschappen te Amsterdam 34, (1931), 372–382.Google Scholar
  63. [63]
    Zagier D.B.: Zetafunktionen und quadratische Körper. Eine Einführung in die höhere Zahlentheorie, springer-verlag, (1981).Google Scholar

Copyright information

© Birkh/:auser Verlag, Basel 1998

Authors and Affiliations

  • Dimitrios I. Dais
    • 1
  • Utz-Uwe Haus
    • 2
  • Martin Henk
    • 2
  1. 1.Mathematisches Institut Universität TübingenTübingenGermany
  2. 2.Konrad-Zuse-Zentrum BerlinBerlinGermany

Personalised recommendations