Exceptional singular $\mathbb{Q}$-homology planes

Palka, Karol

doi:10.5802/aif.2628

Exceptional singular

ℚ

-homology planes [ Plans d’homologie rationnelle exceptionnels et singuliers ]

Palka, Karol

Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 745-774.

Résumé
Abstract

On considère des surfaces $ℚ$ -acycliques singulières dont la partie lisse n’est pas de type général. On démontre que si les singularités sont topologiquement rationnelles, alors soit la partie lisse est réglée par $ℂ^{1}$ ou $ℂ^{*}$ , soit la surface est l’une de deux surfaces exceptionnelles de dimension de Kodaira zéro. Pour les deux surfaces exceptionnelles, la dimension de Kodaira de la partie lisse est zéro, il n’y a qu’un seul point singulier et la singularité est de type $A_{1}$ ou $A_{2}$ , respectivement.

We consider singular $ℚ$ -acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is $ℂ^{1}$ - or $ℂ^{*}$ -ruled or the surface is up to isomorphism one of two exceptional surfaces of Kodaira dimension zero. For both exceptional surfaces the Kodaira dimension of the smooth locus is zero and the singular locus consists of a unique point of type $A_{1}$ and $A_{2}$ respectively.

MR 2895072 | Zbl 1236.14054

DOI : https://doi.org/10.5802/aif.2628
Classification : 14R05, 14J17, 14J26
Mots clés : surface acyclique

@article{AIF_2011__61_2_745_0,
     author = {Palka, Karol},
     title = {Exceptional singular $\mathbb{Q}$-homology planes},
     journal = {Annales de l'Institut Fourier},
     pages = {745--774},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {2},
     year = {2011},
     doi = {10.5802/aif.2628},
     mrnumber = {2895072},
     zbl = {1236.14054},
     language = {en},
     url = {www.numdam.org/item/AIF_2011__61_2_745_0/}
}

Palka, Karol. Exceptional singular $\mathbb{Q}$-homology planes. Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 745-774. doi : 10.5802/aif.2628. http://www.numdam.org/item/AIF_2011__61_2_745_0/

Bibliographie

[1] Abhyankar, Shreeram S. Quasirational singularities, Amer. J. Math., Volume 101 (1979) no. 2, pp. 267-300 | Article | MR 527993 | Zbl 0425.14009

[2] Artebani, Michela; Dolgachev, Igor V. The Hesse pencil of plane cubic curves, arXiv:math/0611590, 2006 | Zbl 1192.14024

[3] Dolgachev, Igor V. Abstract configurations in algebraic geometry, The Fano Conference, pp. 423-462 ((arXiv:math/0304258)) | MR 2112585 | Zbl 1068.14059

[4] Fujita, Takao On the topology of noncomplete algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 29 (1982) no. 3, pp. 503-566 | MR 687591 | Zbl 0513.14018

[5] Gurjar, R. V. Two-dimensional quotients of $C^{n}$ are isomorphic to $C^{2} / Γ$ , Transform. Groups, Volume 12 (2007) no. 1, pp. 117-125 | Article | MR 2308031 | Zbl 1122.32015

[6] Gurjar, R. V.; Miyanishi, Masayoshi Affine lines on logarithmic $Q$ -homology planes, Math. Ann., Volume 294 (1992) no. 3, pp. 463-482 | Article | MR 1188132 | Zbl 0757.14022

[7] Gurjar, R. V.; Pradeep, C. R. $Q$ -homology planes are rational. III, Osaka J. Math., Volume 36 (1999) no. 2, pp. 259-335 | MR 1736480 | Zbl 0954.14013

[8] Iitaka, Shigeru Algebraic geometry, Graduate Texts in Mathematics, Volume 76, Springer-Verlag, New York, 1982 (An introduction to birational geometry of algebraic varieties, North-Holland Mathematical Library, 24) | MR 637060 | Zbl 0491.14006

[9] Kawamata, Yujiro Addition formula of logarithmic Kodaira dimensions for morphisms of relative dimension one, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (1978), pp. 207-217 | MR 578860 | Zbl 0437.14018

[10] Kawamata, Yujiro On the classification of noncomplete algebraic surfaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) (Lecture Notes in Math.) Volume 732, Springer, Berlin, 1979, pp. 215-232 | MR 555700 | Zbl 0407.14012

[11] Kobayashi, Ryoichi Uniformization of complex surfaces, Kähler metric and moduli spaces (Adv. Stud. Pure Math.) Volume 18, Academic Press, Boston, MA, 1990, pp. 313-394 | MR 1145252 | Zbl 0755.32024

[12] Koras, Mariusz A characterization of $A^{2} / Z_{a}$ , Compositio Math., Volume 87 (1993) no. 3, pp. 241-267 | EuDML 90234 | Numdam | MR 1227447 | Zbl 0807.14025

[13] Koras, Mariusz; Russell, Peter Contractible affine surfaces with quotient singularities, Transform. Groups, Volume 12 (2007) no. 2, pp. 293-340 | Article | MR 2323685 | Zbl 1124.14050

[14] Langer, Adrian Logarithmic orbifold Euler numbers of surfaces with applications, Proc. London Math. Soc. (3), Volume 86 (2003) no. 2, pp. 358-396 | Article | MR 1971155 | Zbl 1052.14037

[15] Miyanishi, Masayoshi Open algebraic surfaces, CRM Monograph Series, Volume 12, American Mathematical Society, Providence, RI, 2001 | MR 1800276 | Zbl 0964.14030

[16] Miyanishi, Masayoshi; Sugie, T. Homology planes with quotient singularities, J. Math. Kyoto Univ., Volume 31 (1991) no. 3, pp. 755-788 | MR 1127098 | Zbl 0790.14034

[17] Miyanishi, Masayoshi; Tsunoda, S. Absence of the affine lines on the homology planes of general type, J. Math. Kyoto Univ., Volume 32 (1992) no. 3, pp. 443-450 | MR 1183360 | Zbl 0794.14017

[18] Miyaoka, Yoichi The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann., Volume 268 (1984) no. 2, pp. 159-171 | Article | EuDML 182912 | MR 744605 | Zbl 0521.14013

[20] Palka, Karol Recent progress in the geometry of $Q$ -acyclic surfaces, arXiv:1003.2395, 2010

[21] Pradeep, C. R.; Shastri, Anant R. On rationality of logarithmic $Q$ -homology planes. I, Osaka J. Math., Volume 34 (1997) no. 2, pp. 429-456 | MR 1483859 | Zbl 0890.14021

[22] Tom Dieck, Tammo; Petrie, Ted Homology planes: an announcement and survey, Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988) (Progr. Math.) Volume 80, Birkhäuser Boston, Boston, MA, 1989, pp. 27-48 | MR 1040856 | Zbl 0708.14024

[23] Zaĭdenberg, M. G. Isotrivial families of curves on affine surfaces, and the characterization of the affine plane, Izv. Akad. Nauk SSSR Ser. Mat., Volume 51 (1987) no. 3, p. 534-567, 688 | MR 903623 | Zbl 0666.14018

[24] Zaĭdenberg, M. G. Additions and corrections to the paper: “Isotrivial families of curves on affine surfaces, and the characterization of the affine plane” [Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 534–567; translation in Math. USSR-Izv. 30 (1988), no. 3, 503–532], Izv. Akad. Nauk SSSR Ser. Mat., Volume 55 (1991) no. 2, pp. 444-446 | MR 1133308 | Zbl 0749.14019