Surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree $2$
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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.References
- E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
- Robert D. M. Accola, Topics in the theory of Riemann surfaces, Lecture Notes in Mathematics, vol. 1595, Springer-Verlag, Berlin, 1994. MR 1329541, DOI 10.1007/BFb0073575
- Ingrid Bauer and Fabrizio Catanese, Generic lemniscates of algebraic functions, Math. Ann. 307 (1997), no. 3, 417–444. MR 1437047, DOI 10.1007/s002080050042
- I. Bauer, F. Catanese: Some new surfaces with $p_g=q=0$, preprint AG/0310150v1.
- F. Bardelli and A. Del Centina, Bielliptic curves of genus three: canonical models and moduli space, Indag. Math. (N.S.) 10 (1999), no. 2, 183–190. MR 1816214, DOI 10.1016/S0019-3577(99)80015-9
- W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
- O. Debarre, Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. France 110 (1982), no. 3, 319–346 (French, with English summary). With an appendix by A. Beauville. MR 688038
- Arnaud Beauville, Annulation du $H^1$ et systèmes paracanoniques sur les surfaces, J. Reine Angew. Math. 388 (1988), 149–157 (French). MR 944188, DOI 10.1515/crll.1988.388.149
- Arnaud Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. MR 1406314, DOI 10.1017/CBO9780511623936
- Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
- E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171–219. MR 318163, DOI 10.1007/BF02685880
- G. Borrelli: On the classification of surfaces of general type with non-birational bicanonical map and Du Val double planes, e-print AG/0312351.
- F. Catanese: On a class of surfaces of general type, in Algebraic Surfaces, CIME, Liguori (1981), 269-284.
- F. Catanese, On the moduli spaces of surfaces of general type, J. Differential Geom. 19 (1984), no. 2, 483–515. MR 755236, DOI 10.4310/jdg/1214438689
- Fabrizio Catanese, Singular bidouble covers and the construction of interesting algebraic surfaces, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998) Contemp. Math., vol. 241, Amer. Math. Soc., Providence, RI, 1999, pp. 97–120. MR 1718139, DOI 10.1090/conm/241/03630
- Fabrizio Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math. 122 (2000), no. 1, 1–44. MR 1737256, DOI 10.1353/ajm.2000.0002
- Fabrizio Catanese, Moduli spaces of surfaces and real structures, Ann. of Math. (2) 158 (2003), no. 2, 577–592. MR 2018929, DOI 10.4007/annals.2003.158.577
- F. Catanese and C. Ciliberto, Surfaces with $p_g=q=1$, Problems in the theory of surfaces and their classification (Cortona, 1988) Sympos. Math., XXXII, Academic Press, London, 1991, pp. 49–79. MR 1273372
- F. Catanese and C. Ciliberto, Symmetric products of elliptic curves and surfaces of general type with $p_g=q=1$, J. Algebraic Geom. 2 (1993), no. 3, 389–411. MR 1211993
- Fabrizio Catanese, Ciro Ciliberto, and Margarida Mendes Lopes, On the classification of irregular surfaces of general type with nonbirational bicanonical map, Trans. Amer. Math. Soc. 350 (1998), no. 1, 275–308. MR 1422597, DOI 10.1090/S0002-9947-98-01948-5
- Ciro Ciliberto and Margarida Mendes Lopes, On surfaces with $p_g=q=2$ and non-birational bicanonical maps, Adv. Geom. 2 (2002), no. 3, 281–300. MR 1924760, DOI 10.1515/advg.2002.014
- Ciro Ciliberto, The bicanonical map for surfaces of general type, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 57–84. MR 1492518
- I. Dolgachev, M. Mendes Lopes, and R. Pardini, Rational surfaces with many nodes, Compositio Math. 132 (2002), no. 3, 349–363. MR 1918136, DOI 10.1023/A:1016540925011
- Patrick Du Val, On surfaces whose canonical system is hyperelliptic, Canad. J. Math. 4 (1952), 204–221. MR 48090, DOI 10.4153/cjm-1952-019-3
- Barbara Fantechi and Rita 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, DOI 10.1080/00927879708825927
- P. Francia, On the base points of the bicanonical system, Problems in the theory of surfaces and their classification (Cortona, 1988) Sympos. Math., XXXII, Academic Press, London, 1991, pp. 141–150. MR 1273376
- Robin Hartshorne, Curves with high self-intersection on algebraic surfaces, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 111–125. MR 266924, DOI 10.1007/BF02684600
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- A. Hurwitz: Über Riemann’sche Flachen mit gegeben Verzweigungspunkten, Math. Ann. 39 (1891), 1-61.
- Yoichi Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), no. 2, 159–171. MR 744605, DOI 10.1007/BF01456083
- Margarida Mendes Lopes and Rita Pardini, The bicanonical map of surfaces with $p_g=0$ and $K^2\geq 7$, Bull. London Math. Soc. 33 (2001), no. 3, 265–274. MR 1817764, DOI 10.1017/S0024609301008037
- Margarida Mendes Lopes and Rita Pardini, The bicanonical map of surfaces with $p_g=0$ and $K^2\geq 7$. II, Bull. London Math. Soc. 35 (2003), no. 3, 337–343. MR 1960943, DOI 10.1112/S0024609302001819
- Masayoshi Nagata, On rational surfaces. I. Irreducible curves of arithmetic genus $0$ or $1$, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 32 (1960), 351–370. MR 126443, DOI 10.1215/kjm/1250776405
- E. Palmieri: Gruppi di automorfismi di curve iperellittiche e di rivestimenti ciclici della retta proiettiva, Tesi di Laurea, Università di Roma 2 (2003).
- Rita Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191–213. MR 1103912, DOI 10.1515/crll.1991.417.191
- Rita Pardini, The classification of double planes of general type with $K^2=8$ and $p_g=0$, J. Algebra 259 (2003), no. 1, 95–118. MR 1953710, DOI 10.1016/S0021-8693(02)00539-2
- Igor Reider, Vector bundles of rank $2$ and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), no. 2, 309–316. MR 932299, DOI 10.2307/2007055
- Miles Reid, Campedelli versus Godeaux, Problems in the theory of surfaces and their classification (Cortona, 1988) Sympos. Math., XXXII, Academic Press, London, 1991, pp. 309–365. MR 1273384
- A. Lanteri, M. Palleschi, and D. C. Struppa (eds.), Geometry of complex projective varieties, Seminars and Conferences, vol. 9, Mediterranean Press, Rende, 1993. MR 1225584
- Fernando Serrano, Isotrivial fibred surfaces, Ann. Mat. Pura Appl. (4) 171 (1996), 63–81. MR 1441865, DOI 10.1007/BF01759382
- Gang Xiao, Finitude de l’application bicanonique des surfaces de type général, Bull. Soc. Math. France 113 (1985), no. 1, 23–51 (French, with English summary). MR 807825, DOI 10.24033/bsmf.2019
- Gang Xiao, Surfaces fibrées en courbes de genre deux, Lecture Notes in Mathematics, vol. 1137, Springer-Verlag, Berlin, 1985 (French). MR 872271, DOI 10.1007/BFb0075351
- Gang Xiao, Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), no. 3, 449–466. MR 875340, DOI 10.1007/BF01450841
- Gang Xiao, Degree of the bicanonical map of a surface of general type, Amer. J. Math. 112 (1990), no. 5, 713–736. MR 1073006, DOI 10.2307/2374804
Additional Information
- Francesco Polizzi
- Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy
- MR Author ID: 723443
- Email: polizzi@mat.uniroma2.it, polizzi@math.jussieu.fr
- Received by editor(s): November 27, 2003
- Received by editor(s) in revised form: March 10, 2004
- Published electronically: March 25, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 759-798
- MSC (2000): Primary 14J29, 14J10, 14H37
- DOI: https://doi.org/10.1090/S0002-9947-05-03673-1
- MathSciNet review: 2177040