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Numerical Godeaux surfaces with an involution

Authors: Alberto Calabri, Ciro Ciliberto and Margarida Mendes Lopes
Journal: Trans. Amer. Math. Soc. 359 (2007), 1605-1632
MSC (2000): Primary 14J29
Published electronically: October 17, 2006
MathSciNet review: 2272143
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Abstract: Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and $ K^2=1$ and are usually called numerical Godeaux surfaces. Although they have been studied by several authors, their complete classification is not known.

In this paper we classify numerical Godeaux surfaces with an involution, i.e. an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested by Campedelli, or is the union of two lines and a curve of degree 12 with certain singularities. The latter type of double planes are degenerations of examples described by Du Val, and their existence was previously unknown; we show some examples of this new type, also computing their torsion group.

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  • [AS] M.F. Atiyah, I.M. Singer, The index of elliptic operators: III, Ann. of Math. 87 (1968), 546-604.MR 0236952 (38:5245)
  • [Bar1] R. Barlow, Some new surfaces with $ p\sb g=0$, Duke Math. J. 51 (1984), no. 4, 889-904. MR 0771386 (86c:14032)
  • [Bar2] R. Barlow, A simply connected surface of general type with $ p\sb g=0$, Invent. Math. 79 (1985), no. 2, 293-301. MR 0778128 (87a:14033)
  • [Be] A. Beauville, Sur le nombre maximum de points doubles d'une surface dans $ \mathbf{P}^3 (\mu (5)=31)$, in ``Journées de Géométrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry'', Angers, 1979, pp. 207-215, Sijthoff & Noordhoff, 1980.MR 0605342 (82k:14037)
  • [BPV] W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 4, Springer, 1984. MR 0749574 (86c:32026)
  • [Bl] S. Bloch, Lectures on algebraic cycles, Lecture one, Duke University Math. Series IV, Durham, 1980. MR 0558224 (82e:14012)
  • [Bo] G. Borrelli, On the classification of surfaces of general type with non-birational bicanonical map and Du Val double planes, preprint, math.AG n. 0312351.
  • [CCM] A. Calabri, C. Ciliberto, M. Mendes Lopes, Rational surfaces with an even set of four nodes, Math. Res. Lett. 11 (2004), n. 6, 799-808. MR 2099361 (2006c:14061)
  • [CF] A. Calabri, R. Ferraro, Explicit resolutions of double points singularities of surfaces, Collect. Math. 53 (2002), n. 2, 99-131.MR 1913513 (2003h:32043)
  • [Cam] L. Campedelli, Sopra alcuni piani doppi notevoli con curva di diramazione del decimo ordine, Atti Accad. Naz. Lincei 15 (1932), 536-542.
  • [CaD] F. Catanese, O. Debarre, Surfaces with $ K\sp 2=2$, $ p\sb g=1$, $ q=0$, J. Reine Angew. Math. 395 (1989), 1-55. MR 0983058 (89m:14017)
  • [CL] F. Catanese, C. LeBrun, On the scalar curvature of Einstein manifolds, Math. Res. Lett. 4 (1997), no. 6, 843-854. MR 1492124 (98k:53057)
  • [CP] F. Catanese, R. Pignatelli, On simply connected Godeaux surfaces, in T. Peternell, F.-O. Schreyer (eds.), Complex analysis and algebraic geometry. A volume in memory of Michael Schneider, 117-153, de Gruyter, 2000.MR 1760875 (2001g:14064)
  • [Ci1] C. Ciliberto, The bicanonical map for surfaces of general type, in Algebraic geometry--Santa Cruz 1995, 57-84, Proc. Sympos. Pure Math. 62, Part 1, Amer. Math. Soc. 1997.MR 1492518 (98m:14040)
  • [Ci2] C. Ciliberto, The Geometry of Algebraic Varieties, in Development of Mathematics 1950-2000, edited by J.-P. Pier, Birkhäuser-Verlag, 2000.MR 1796844 (2002k:14001)
  • [CoD] F. Cossec, I. Dolgachev, Enriques surfaces I, Progress in Math. 76, Birkhäuser, 1989.MR 0986969 (90h:14052)
  • [CG] P.C. Craighero, R. Gattazzo, Quintic surfaces of $ \mathbb{P}^3$ having a nonsingular model with $ q=p\sb g=0$, $ P\sb 2\not=0$, Rend. Sem. Mat. Univ. Padova 91 (1994), 187-198. MR 1289636 (95g:14038)
  • [Do] I. Dolgachev, On algebraic surfaces with $ q=p_g=0$, in G. Tomassini (ed.), Algebraic surfaces, III ciclo 1977, Villa Monastero-Varenna, Como. C.I.M.E., Liguori, 1988.
  • [DMP] I. Dolgachev, M. Mendes Lopes, R. Pardini, Rational surfaces with many nodes, Compositio Math. 132 (2002), no. 3, 349-363.MR 1918136 (2003g:14049)
  • [DW] I. Dolgachev, C. Werner, A simply connected numerical Godeaux surface with ample canonical class, J. Algebraic Geom. 8 (1999), 737-764, with Erratum in J. Algebraic Geom. 10 (2001), 397. MR 1703612 (2000h:14030); MR 1811560 (2002a:14041)
  • [Du] P. Du Val, On surfaces whose canonical system is hyperelliptic, Canadian J. of Math. 4 (1952), 204-221. MR 0048090 (13:977c)
  • [EV] H. Esnault, E. Viehweg, Lectures on vanishing theorems, DMV Seminar, Band 20, Birkhäuser, 1992. MR 1193913 (94a:14017)
  • [Fr] M. Freedman, The topology of 4-manifolds, J. Differential Geometry 17 (1982), 357-454.MR 0679066 (84b:57006)
  • [Go1] L. Godeaux, Sur une surface algébriques de genre zero et de bigenre deux, Atti Accad. Naz. Lincei 14 (1931), 479-481.
  • [Go2] L. Godeaux, Les surfaces algébriques non rationelles de genre arithmétique et géométriques nuls, Actulité Scientifiques et Industrielles 123, Exposés de Geométrie, IV, Hermann, Paris, 1934.
  • [Gr] M. Greenberg, Algebraic Topology: A First Course, W. A. Benjamin Publ., Reading, Mass., 1981.MR 0643101 (83b:55001)
  • [Ho] E. Horikawa, On algebraic surfaces with pencils of curves of genus $ 2$, in Complex analysis and algebraic geometry, 79-90, Iwanami Shoten, 1977. MR 0453756 (56:12015)
  • [Ke] J.H. Keum, Some new surfaces of general type with $ p_g=0$, preprint, 1988.
  • [KL] J.H. Keum, Y.  Lee, Fixed locus of an involution acting on a Godeaux surface, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 2, 205-216.MR 1765910 (2001f:14074)
  • [Ma] Maple 7, Waterloo Maple Inc., Waterloo, Ontario.
  • [Me] M. Mendes Lopes, Adjoint systems on surfaces, Boll. Un. Mat. Ital. A (7) 10 (1996), no. 1, 169-179. MR 1386254 (97d:14011)
  • [MP1] M. Mendes Lopes, R. Pardini, The bicanonical map of surfaces with $ p_g=0$ and $ K^2\ge 7$, Bull. London Math. Soc. 33 (2001), 1-10.MR 1817764 (2002a:14042)
  • [MP2] M. Mendes Lopes, R. Pardini, A connected component of the moduli space of surfaces of general type with $ p_g=0$, Topology 40 (2001), no. 5, 977-991.MR 1860538 (2002g:14052)
  • [Mi] Y. Miyaoka, Tricanonical maps of numerical Godeaux surfaces, Invent. Math. 34 (1976), no. 2, 99-111.MR 0409481 (53:13236)
  • [Mu] M. Murakami, The torsion group of a certain numerical Godeaux surface, J. Math. Kyoto Univ. 41 (2001), no. 2, 323-333. MR 1852987 (2002i:14040)
  • [Na] D. Naie, Surfaces d'Enriques et une construction de surfaces de type général avec $ p_g = 0$, Math. Z. 215 (2) (1994), 269-280.MR 1259462 (94m:14055)
  • [Re1] M. Reid, Surfaces with $ p\sb{g}=0$, $ K\sp{2}=1$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1978), no. 1, 75-92. MR 0494596 (80h:14018)
  • [Re2] M. Reid, Campedelli versus Godeaux, in Problems in the theory of surfaces and their classification (Cortona, 1988), 309-365, Sympos. Math. 32, Academic Press, 1991.MR 1273384 (95h:14031)
  • [OP] F. Oort, C. Peters, A Campedelli surface with torsion group $ Z/2$, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 4, 399-407.MR 0639857 (83h:14032)
  • [St] E. Stagnaro, On Campedelli branch loci, Ann. Univ. Ferrara Sez. VII (N.S.) 43 (1997), 1-26.MR 1686746 (2000e:14061)
  • [We1] C. Werner, A surface of general type with $ p\sb g=q=0$, $ K\sp 2=1$, Manuscripta Math. 84 (1994), no. 3-4, 327-341. MR 1291124 (95g:14042)
  • [We2] C. Werner, A four-dimensional deformation of a numerical Godeaux surface, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1515-1525.MR 1407503 (97k:14034)
  • [We3] C. Werner, Branch curves for Campedelli double planes, to appear in Rocky Mountain J. Math.
  • [Xi] G. Xiao, Surfaces fibrées en courbes de genre deux, Lecture Notes in Mathematics 1137, Springer, 1985. MR 0872271 (88a:14042)

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Additional Information

Alberto Calabri
Affiliation: Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Padova, via Trieste 63, I-35131 Padova, Italy

Ciro Ciliberto
Affiliation: Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy

Margarida Mendes Lopes
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Keywords: Godeaux surface, involution, torsion group
Received by editor(s): January 19, 2005
Published electronically: October 17, 2006
Additional Notes: This research has been carried out in the framework of the EAGER project financed by the EC, project n. HPRN-CT-2000-00099. The first two authors are members of G.N.S.A.G.A.-I.N.d.A.M., which generously supported this research. The third author is a member of the Center for Mathematical Analysis, Geometry and Dynamical Systems, IST, and was partially supported by FCT (Portugal) through program POCTI/FEDER and Project POCTI/MAT/44068/2002.
Article copyright: © Copyright 2006 American Mathematical Society

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