Contraction of excess fibres between the McKay correspondences in dimensions two and three

Boissière, Samuel; Sarti, Alessandra

Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 1839-1861.

Les singularités quotients de dimensions deux et trois obtenues par des groupes polyédraux et les groupes polyédraux binaires correspondants admettent des résolutions de singularités naturelles par les schémas de Hilbert d’orbites régulières, dont les fibres exceptionnelles au-dessus de l’origine révèlent des propriétés similaires. Nous construisons un morphisme entre ces deux résolutions, contractant exactement la partie excédentaire de la fibre exceptionnelle. Cette construction est motivée par l’étude de certains pinceaux de surfaces K3 apparaissant comme résolutions minimales de quotients de surfaces nodales à grandes symétries.

The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres over the origin reveal similar properties. We construct a morphism between these two resolutions, contracting exactly the excess part of the exceptional fibre. This construction is motivated by the study of some pencils of K3 surfaces appearing as minimal resolutions of quotients of nodal surfaces with high symmetries.

Classification : 14C05, 14E15, 20C15, 51F15
Keywords: Quotient singularities, McKay correspondence, Hilbert schemes, polyhedral groups
Mot clés : singularités quotients, correspondance de McKay, schémas de Hilbert, groupes polyédraux

[1] Atiyah, M. F.; Macdonald, I. G. Introduction to commutative algebra, Addison-Wesley, 1969 | MR | Zbl

[2] Barth, W. P.; Sarti, A. Polyhedral Groups and Pencils of K3-Surfaces with Maximal Picard Number, Asian J. of Math., Volume 7 (2003) no. 4, pp. 519-538 | MR | Zbl

[3] Bridgeland, T.; King, A.; Reid, M. The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc., Volume 14 (2001) no. 3, pp. 535-554 | DOI | MR | Zbl

[4] Crawley-Boevey, W. On the exceptional fibres of Kleinian singularities, Amer. J. Math., Volume 122 (2000), pp. 1027-1037 | DOI | MR | Zbl

[5] Fogarty, J. Algebraic families on an algebraic surface, Amer. J. Math., Volume 90 (1968), pp. 511-521 | DOI | MR | Zbl

[6] Gomi, Y.; Nakamura, I.; Shinoda, K.-I. Hilbert schemes of $G$-orbits in dimension three, Asian J. Math., Volume 4 (2000), pp. 51-70 | MR | Zbl

[7] Gomi, Y.; Nakamura, I.; Shinoda, K.-I. Coinvariant algebras of finite subgroups of $SL(3,ℂ)$, Can. J. Math., Volume 56 (2002), pp. 495-528 | DOI | MR | Zbl

[8] Gonzalez-Sprinberg, G.; Verdier, J.-L. Construction géométrique de la correspondance de McKay, Ann. scient. Éc. Norm. Sup., Volume 16 (1983), pp. 409-449 | Numdam | MR | Zbl

[9] Hartshorne, R. Algebraic geometry, Springer, 1977 | MR | Zbl

[10] Huybrechts, D.; Lehn, M. The geometry of moduli spaces of sheaves, Vieweg, 1997 | MR | Zbl

[11] Ito, Y.; Nakajima, H. McKay correspondence and Hilbert schemes in dimension $3$, Topology, Volume 39 (2000), pp. 1155-1191 | DOI | MR | Zbl

[12] Ito, Y.; Nakamura, I. McKay correspondence and Hilbert schemes, Proc. Japan. Acad., Volume 92 (1996), pp. 135-138 | MR | Zbl

[13] Ito, Y.; Nakamura, I. Hilbert schemes and simple singularities, New trends in algebraic geometry, Camb. Univ. Press, July 1996 Warwick European alg. geom. conf. (1999), pp. 151-233 | MR | Zbl

[14] Kapranov, M.; Vasserot, E. Kleinian singularities, derived categories and Hall algebras, Math. Ann., Volume 316 (2000), pp. 565-576 | DOI | MR | Zbl

[15] McKay, J. Graphs, singularities and finite groups, Proc. of Symp. in Pure Math., Volume 37 (1980), pp. 183-186 | MR | Zbl

[16] Nakamura, I. Hilbert scheme of abelian group orbits, J. Alg. Geom., Volume 10 (2001), pp. 757-779 | MR | Zbl

[17] Sarti, A. Pencils of Symmetric Surfaces in ${\mathbb{P}}_3$, J. of Algebra, Volume 246 (2001), pp. 429-452 | DOI | MR | Zbl

[18] Térouanne, S. Correspondance de McKay : variations en dimension trois, Université Joseph Fourier de Grenoble (2004) (Ph. D. Thesis)