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Transactions of the American Mathematical Society

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Surfaces of general type with $p_g=q=1, \; K^2=8$ and bicanonical map of degree $2$

Author: Francesco Polizzi
Journal: Trans. Amer. Math. Soc. 358 (2006), 759-798
MSC (2000): Primary 14J29, 14J10, 14H37
Published electronically: March 25, 2005
MathSciNet review: 2177040
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Abstract: We classify the minimal algebraic surfaces of general type with $p_g=q=1, \; K^2=8$ and bicanonical map of degree $2$. It will turn out that they are isogenous to a product of curves, i.e. if $S$ is such a surface, then there exist two smooth curves $C, \; F$ and a finite group $G$ acting freely on $C \times F$ such that $S = (C \times F)/G$. We describe the $C, \; F$ and $G$that occur. In particular the curve $C$ is a hyperelliptic-bielliptic curve of genus $3$, and the bicanonical map $\phi$ of $S$ is composed with the involution $\sigma$ induced on $S$ by $\tau \times id: C \times F \longrightarrow C \times F$, where $\tau$ is the hyperelliptic involution of $C$. In this way we obtain three families of surfaces with $p_g=q=1, \; K^2=8$which yield the first-known examples of surfaces with these invariants. We compute their dimension and we show that they are three generically smooth, irreducible components of the moduli space $\mathcal{M}$ of surfaces with $p_g=q=1, \; K^2=8$. Moreover, we give an alternative description of these surfaces as double covers of the plane, recovering a construction proposed by Du Val.

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  • [ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris: Geometry of algebraic curves, Springer-Verlag 1985. MR 0770932 (86h:14019)
  • [Ac94] R. Accola: Topics in the theory of the Riemann surfaces, Lecture Notes in Mathematics 1595 (1994). MR 1329541 (97k:30053)
  • [BaCa97] I. Bauer, F. Catanese: Generic lemniscates of algebraic functions, Math. Ann. 307 (1997), 417-444. MR 1437047 (98f:57003)
  • [BaCa03] I. Bauer, F. Catanese: Some new surfaces with $p_g=q=0$, preprint AG/0310150v1.
  • [BaDC99] F. Bardelli, A. Del Centina: Bielliptic curves of genus three: canonical models and moduli spaces, Indag. Mathem., N.S., 10 (2) (1999), 183-190. MR 1816214 (2001m:14039)
  • [BPV84] W. Barth, C. Peters, A. Van de Ven: Compact Complex Surfaces, Springer-Verlag 1984. MR 0749574 (86c:32026)
  • [Be82] A. Beauville: L'inegalité $p_g \geq 2q-4$ pour les surfaces de type générale, Bull. Soc. Math. de France 110 (1982), 343-346. MR 0688038 (84f:14026)
  • [Be88] A. Beauville: Annulation du $H^1$ et systèmes paracanoniques sur le surfaces, J. Reine Angew. Math. 388 (1988), 149-157. MR 0944188 (89i:14032)
  • [Be96] A. Beauville: Complex algebraic surfaces, Cambridge University Press 1996. MR 1406314 (97e:14045)
  • [Bir74] J. Birman: Braid, links and mapping class groups, Annals of Mathematics Studies 82, Princeton University Press 1974. MR 0375281 (51:11477)
  • [Bo73] E. Bombieri: Canonical models of surfaces of general type, Publ. IHES 42 (1973), 171-219. MR 0318163 (47:6710)
  • [Bor03] G. Borrelli: On the classification of surfaces of general type with non-birational bicanonical map and Du Val double planes, e-print AG/0312351.
  • [Ca81] F. Catanese: On a class of surfaces of general type, in Algebraic Surfaces, CIME, Liguori (1981), 269-284.
  • [Ca84] F. Catanese: On the moduli space of surfaces of general type, Journal of Differential Geometry 19 (1984), no. 2, 483-515. MR 0755236 (86h:14031)
  • [Ca99] F. Catanese: Singular bidouble covers and the construction of interesting algebraic surfaces, Contemporary Mathematics 241 (1999), 97-119. MR 1718139 (2000j:14061)
  • [Ca00] F. Catanese: Fibred surfaces, varieties isogenous to a product and related moduli spaces, American Journal of Mathematics 122 (2000), 1-44. MR 1737256 (2001i:14048)
  • [Ca02] F. Catanese: Moduli spaces of surfaces and real structures, Annals of Mathematics 158, n.2 (2003), 577-592. MR 2018929
  • [CaCi91] F. Catanese and C. Ciliberto: Surfaces with $p_g=q=1$, Symposia Math. 32 (1991), 49-79. MR 1273372 (95d:14030)
  • [CaCi93] F. Catanese and C. Ciliberto: Symmetric product of elliptic curves and surfaces of general type with $p_g=q=1$, J. of Algebraic Geometry 2 (1993), 389-411. MR 1211993 (94i:14040)
  • [CCM98] F. Catanese, C. Ciliberto and M. M. Lopes: Of the classification of irregular surfaces of general type with non birational bicanonical map, Trans. of the Amer. Math. Soc. 350 (1998), 275-308.MR 1422597 (98h:14043)
  • [CM02] C. Ciliberto, M. Mendes Lopes: On surfaces with $p_g=q=2$and non-birational bicanonical map, Adv. Geom. 2 (2002), no. 3, 281-300. MR 1924760 (2004d:14053)
  • [Ci97] C. Ciliberto: The bicanonical map for surfaces of general type, Proc. of Symp. in Pure Mathematics 62.1 (1997), 57-84.MR 1492518 (98m:14040)
  • [DMP02] I. Dolgachev, M. Mendes Lopes, R. Pardini: Rational surfaces with many nodes, Compositio Math. 132 (2002), no. 3, 349-363. MR 1918136 (2003g:14049)
  • [DV52] P. Du Val, On surfaces whose canonical system is hyperelliptic, Canadian J. of Math. 4 (1952), 204-221. MR 0048090 (13:977c)
  • [FP97] B. Fantechi, R. Pardini: Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers, Comm. Algebra 25 (1997), no. 5, 1413-1441. MR 1444010 (98c:14028)
  • [Fr91] P. Francia: On the base points of the bicanonical system, Symposia Math. 32 (1991), 141-150. MR 1273376 (95a:14004)
  • [Ha69] R. Hartshorne: Curves with high self-intersection on algebraic surfaces, Publ. Math. IHES 36 (1969), 111-125. MR 0266924 (42:1826)
  • [Ha77] R. Hartshorne: Algebraic Geometry, Lecture Notes in Mathematics 52, Springer-Verlag 1977. MR 0463157 (57:3116)
  • [Hur891] A. Hurwitz: Über Riemann'sche Flachen mit gegeben Verzweigungspunkten, Math. Ann. 39 (1891), 1-61.
  • [Mi84] Y. Miyaoka: The maximum number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), 159-171. MR 0744605 (85j:14060)
  • [MP01] M. Mendes Lopes, R. Pardini: The bicanonical map of surfaces with $p_g=0$ and $K^2=7$, Bull. London Math. Soc. 33 no. 3 (2001), 265-274. MR 1817764 (2002a:14042)
  • [MP03] M. Mendes Lopes, R. Pardini: The bicanonical map of surfaces with $p_g=0$ and $K^2=7$, II. Bull. London Math. Soc. 35 no. 3 (2003), 337-343. MR 1960943 (2004a:14040)
  • [Na60] M. Nagata: On rational surfaces I, Mem. Coll. Sci., U. of Kyoto Ser A 32 (1960), 351-370. MR 0126443 (23:A3739)
  • [Pal03] E. Palmieri: Gruppi di automorfismi di curve iperellittiche e di rivestimenti ciclici della retta proiettiva, Tesi di Laurea, Università di Roma 2 (2003).
  • [Par91] R. Pardini: Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191-213. MR 1103912 (92g:14012)
  • [Par03] R. Pardini: The classification of double planes of general type with $K^2=8$ and $p_g=0$, Journal of Algebra 259 (2003) no. 3, 95-118. MR 1953710 (2004a:14041)
  • [Re88] I. Reider: Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. 127 (1988), 309-316. MR 0932299 (89e:14038)
  • [Reid91] M. Reid: Campedelli versus Godeaux, Symposia Math. 32 (1991), 309-365. MR 1273384 (95h:14031)
  • [Se90] F. Serrano: Fibrations on algebraic surfaces, Geometry of Complex Projective Varieties (Cetraro 1990), A. Lanteri, M. Palleschi, D. C. Struppa eds., Mediterranean Press (1993), 291-300. MR 1225584 (93m:14001)
  • [Se96] F. Serrano: Isotrivial fibred surfaces, Annali di Matematica pura e applicata, vol. CLXXI (1996), 63-81. MR 1441865 (98e:14036)
  • [Xi85a] G. Xiao: Finitude de l' application bicanonique des surfaces de type générale, Boll. Soc. Math. de France 113 (1985), 23-51. MR 0807825 (87a:14035)
  • [Xi85b] G. Xiao: Surfaces fibrées en courbes de genre deux, Lecture Notes in Mathematics 1137 (1985). MR 0872271 (88a:14042)
  • [Xi87] G. Xiao: Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), 449-466. MR 0875340 (88a:14046)
  • [Xi90] G. Xiao: Degree of the bicanonical map of a surface of general type, Amer. J. of Math. 112 (5), (1990) 309-316. MR 1073006 (91i:14030)

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Additional Information

Francesco Polizzi
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy

Keywords: Surfaces of general type, bicanonical map, isotrivial fibrations, Galois coverings
Received by editor(s): November 27, 2003
Received by editor(s) in revised form: March 10, 2004
Published electronically: March 25, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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