Exceptional singular -homology planes
Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 745-774.

We consider singular -acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is 1 - or * -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.

On considère des surfaces -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 1 ou * , 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.

DOI: 10.5802/aif.2628
Classification: 14R05, 14J17, 14J26
Keywords: Acyclic surface, homology plane, exceptional Q-homology plane
Mot clés : surface acyclique
Palka, Karol 1, 2

1 University of Warsaw Institute of Mathematics ul. Banacha 2 02-097 Warsaw (Poland)
2 Institute of Mathematics Polish Academy of Sciences ul. Śniadeckich 8 00-956 Warsaw (Poland)
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Palka, Karol. Exceptional singular $\mathbb{Q}$-homology planes. Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 745-774. doi : 10.5802/aif.2628. http://archive.numdam.org/articles/10.5802/aif.2628/

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

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

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

[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 | Zbl

[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 | DOI | MR | Zbl

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

[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 | Zbl

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

[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), Kinokuniya Book Store, Tokyo (1978), pp. 207-217 | MR | Zbl

[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 | Zbl

[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 | Zbl

[12] Koras, Mariusz A characterization of A 2 /Z a , Compositio Math., Volume 87 (1993) no. 3, pp. 241-267 | EuDML | Numdam | MR | Zbl

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

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

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

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

[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 | Zbl

[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 | DOI | EuDML | MR | Zbl

[19] Mumford, David The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. (1961) no. 9, pp. 5-22 | DOI | EuDML | Numdam | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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