Affine normal surfaces with simply-connected smooth locus. (English) Zbl 1260.14057
In the article under review, the authors investigate algebraic and topological conditions which imply that a given normal complex affine surface is smooth or has at worst rational singularities. They establish in particular a kind of global counterpart to Mumford’s topological criterion for smoothness of germs of complex surfaces: a normal complex affine surface is smooth if it is topologically contractible and its smooth locus is simply connected. A an algebro-topological characterization of the affine plane among surfaces with quasi-rational singularities is also given.
Reviewer: Adrien Dubouloz (Dijon)
MSC:
| 14L30 | Group actions on varieties or schemes (quotients) |
| 14R20 | Group actions on affine varieties |
References:
| [1] | Abhyankar S.S.: Quasirational singularities. Am. J. Math. 101(2), 267–300 (1979) · Zbl 0425.14009 · doi:10.2307/2373979 |
| [2] | Abyankar S.S., Moh T.T.: Embeddings of the line in the plane. J. Reine Angew. Math. 276, 148–166 (1975) · Zbl 0332.14004 |
| [3] | Brieskorn, E.: Rationale singularitaten komplexer Flachen. Invent. Math. 4, 336–358 (1967/1968) · Zbl 0219.14003 |
| [4] | Bundagaard, S., Nielsen, J.: On normal subgroups with finite index in F-groups. Mat. Tidsskr. B 32–53 (1951) |
| [5] | Fujita T.: On the topology of noncomplete algebraic surfaces. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29(3), 503–566 (1982) · Zbl 0513.14018 |
| [6] | Gurjar R.V.: Two-dimensional quotients of $${\(\backslash\)mathbb{C}\^n}$$ are isomorphic to $${\(\backslash\)mathbb{C}\^2/\(\backslash\)Gamma}$$ . Transf. Groups 12(1), 117–125 (2007) · Zbl 1122.32015 · doi:10.1007/s00031-005-1129-y |
| [7] | Gurjar R.V., Miyanishi M.: Automorphisms of affine surfaces with $${\(\backslash\)mathbb{A}\^1}$$ -fibrations. Michigan Math. J. 53, 33–55 (2005) · Zbl 1084.14062 · doi:10.1307/mmj/1114021083 |
| [8] | Gurjar R.V., Pradeep C.R.: $${\(\backslash\)mathbb{Q}}$$ -homology planes are rational-III. Osaka J. Math. 36, 259–335 (1999) · Zbl 0954.14013 |
| [9] | Kaliman Sh., Saveliev N.: $${\(\backslash\)mathbb{C}_+}$$ -actions on contractible threefolds. Michigan Math. J. 52, 619–625 (2004) · Zbl 1067.14067 · doi:10.1307/mmj/1100623416 |
| [10] | Kawamata, Y.: On the classification of non-complete algebraic surfaces. In: Algebraic Geometry. Lecture Notes in Mathematics, vol. 732, pp. 215–232. Springer, Berlin (1979) · Zbl 0407.14012 |
| [11] | Kobayashi, R.: Uniformization of complex surfaces. In: Kahler metric and moduli spaces. Adv. Stud. Pure Math., vol. 18-II, pp. 313–394. Academic Press, Boston (1990) · Zbl 0755.32024 |
| [12] | Koras M., Russell P.: Contractible affine surfaces with a quotient singularity. Transf. Groups 12(2), 293–340 (2007) · Zbl 1124.14050 · doi:10.1007/s00031-006-0036-z |
| [13] | Kraft H.P., Petrie T., Randall J.D.: Quotient varieties. Adv. Math. 74(2), 145–162 (1989) · Zbl 0691.14029 · doi:10.1016/0001-8708(89)90007-8 |
| [14] | Miyanishi, M.: Open algebraic surfaces. In: CRM Monograph Series, vol. 12. American Mathematical Society, Providence (2001) · Zbl 0964.14030 |
| [15] | Miyanishi, M.: Normal affine subalgebras of a polynomial ring. In: Algebraic and Topological Theories, to the memory of Dr. Takehiko MIYATA, Kinokuniya, pp. 37–51 (1985) |
| [16] | Miyanishi M.: An algebric characterization of the affine plane. J. Math. Kyoto Univ. 15, 169–184 (1975) · Zbl 0304.14021 |
| [17] | Miyanishi M., Sugie T.: Homology planes with quotient singularities. J. Math. Kyoto Univ. 31(3), 755–788 (1991) · Zbl 0790.14034 |
| [18] | Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes Etudes Sci. Publ. Math., vol. 9, pp. 5–22 (1961) · Zbl 0108.16801 |
| [19] | Nori M.V.: Zariski’s conjecture and related problems. Ann. Sci. Ecole Norm. Sup. (4) 16(2), 305–344 (1983) · Zbl 0527.14016 |
| [20] | Palka, K.: Singular $${\(\backslash\)mathbb{Q}}$$ -homology planes I. arXiv:0806.3110 · Zbl 1236.14054 |
| [21] | Zariski O.: Interprétations algébrico-géométriques du quatorzième problème de Hilbert. Bull. Sci. Math. 78, 155–168 (1954) · Zbl 0056.39602 |
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