Contraction of excess fibres between the McKay correspondences in dimensions two and three
Annales de l'Institut Fourier, Volume 57 (2007) no. 6, p. 1839-1861

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.

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.

DOI : https://doi.org/10.5802/aif.2315
Classification:  14C05,  14E15,  20C15,  51F15
Keywords: Quotient singularities, McKay correspondence, Hilbert schemes, polyhedral groups
@article{AIF_2007__57_6_1839_0,
     author = {Boissi\`ere, Samuel and Sarti, Alessandra},
     title = {Contraction of excess fibres between the McKay correspondences in dimensions two and three},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {6},
     year = {2007},
     pages = {1839-1861},
     doi = {10.5802/aif.2315},
     mrnumber = {2377888},
     zbl = {1133.14004},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2007__57_6_1839_0}
}
Boissière, Samuel; Sarti, Alessandra. Contraction of excess fibres between the McKay correspondences in dimensions two and three. Annales de l'Institut Fourier, Volume 57 (2007) no. 6, pp. 1839-1861. doi : 10.5802/aif.2315. http://www.numdam.org/item/AIF_2007__57_6_1839_0/

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