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Transactions of the American Mathematical Society

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On the classification of irregular surfaces
of general type with
nonbirational bicanonical map


Authors: Fabrizio Catanese, Ciro Ciliberto and Margarida Mendes Lopes
Journal: Trans. Amer. Math. Soc. 350 (1998), 275-308
MSC (1991): Primary 14J29
DOI: https://doi.org/10.1090/S0002-9947-98-01948-5
MathSciNet review: 1422597
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Abstract: The present paper is devoted to the classification of irregular surfaces of general type with $p_{g}\geq 3$ and nonbirational bicanonical map. Our main result is that, if $S$ is such a surface and if $S$ is minimal with no pencil of curves of genus $2$, then $S$ is the symmetric product of a curve of genus $3$, and therefore $p_{g}=q=3$ and $K^{2}=6$. Furthermore we obtain some results towards the classification of minimal surfaces with $p_{g}=q=3$. Such surfaces have $6\leq K^{2}\leq 9$, and we show that $K^{2}=6$ if and only if $S$ is the symmetric product of a curve of genus $3$. We also classify the minimal surfaces with $p_{g}=q=3$ with a pencil of curves of genus $2$, proving in particular that for those one has $K^{2}=8$.


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  • [B] E. Bombieri, Canonical models of surfaces of general type, Publ. Math. IHES, 42 (1973), 171-219. MR 47:6710
  • [Be1] A. Beauville, L'inégalité $p_{g}\geq 2q-4$ pour les surfaces de type générale, Appendix to [De1], Bull. Soc. Math. France, 110 (1982), 343-346. MR 84f:14026
  • [Be2] A. Beauville, Annulation du $H^{1}$ et systémes paracanoniques sur les surfaces J. Reine Angew. Math. 388 (1988), 149-157. MR 81i:14032
  • [BPV] W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces, Ergebnisse der Math. (3) 4, Springer-Verlag, 1984. MR 86c:32026
  • [BS] M. Beltrametti, A. Sommese, Zero-cycles and kth order embeddings of smooth projective surfaces, in Problems in the Theory of Surfaces and Their Classification (F. Catanese, C. Ciliberto, M. Cornalba, ed.), Symposia Math. (INDAM) 32 (1991), 33-48. MR 95d:14005
  • [Ca] F. Catanese, On the moduli spaces of surfaces of general type, J. Differential Geometry, 19 (1984), 483-515. MR 86h:14031
  • [CFM] C. Ciliberto, P. Francia, M. Mendes Lopes, Remarks on the bicanonical map for surfaces of general type, Math.Z. 224 (1997), 137-166.
  • [D] P. Du Val, On surfaces whose canonical system is hyperelliptic, Canadian J. Math., 4 (1952), 204-221. MR 13:977e
  • [De1] O. Debarre, Inégalités numériques pour les surfaces de type general, Bull. Soc. Math. de France, 110 (1982), 319-346. MR 84f:14026
  • [De2] O. Debarre, Théorèmes de connexité et variétés abéliennes, Amer. J. Math 117 (1995), 787-805. MR 96j:14009
  • [GL] M. Green, R. Lazarsfeld, Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 4 (1991), 87-103. MR 92i:32021
  • [Ho] E. Horikawa, Algebraic surfaces of general type with small $c_{1}^{2}$, J. Fac. Sci. Univ. Tokyo, Sec. IA, Math. 28 (1981), 745-755.MR 84d:14019
  • [M] M. Mendes Lopes, Adjoint systems on surfaces, Boll. Unione Mat. Italiana A(7) 10 (1996), 169-179. CMP 96:11
  • [Mi] Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), 159-171. MR 85j:14060
  • [P] G. Pirola, Curves on Kummer varieties, pre-print.
  • [Ra] C. P. Ramanujam, Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. 36 (1972), 41-51, suppl. ibidem 38 (1974), 121-124. MR 48:8502; MR 52:13859
  • [R] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. 127 (1988), 309-316. MR 89e:14038
  • [X1] G. Xiao, Degree of the bicanonical map of a surface of general type, Amer. J. Math., 112 (5) (1990), 713-737. MR 91i:14030
  • [X2] G. Xiao, L'irrégularité des surfaces de type général dont le systéme canonique est composé d'un pinceau, Compositio Math. 56 (1985), 251-257. MR 87d:14031
  • [X3] G. Xiao, Irregularity of surfaces with a linear pencil, Duke Math. J., 55 (1987), 597-602. MR 89c:14055

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Additional Information

Fabrizio Catanese
Affiliation: Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy
Address at time of publication: Mathematisches Institut der Georg-August, Universität Göttingen, Bunsenstraße 3-5, D-37073 Göttingen, Germany
Email: catanese@uni-math.gwdg.de

Ciro Ciliberto
Affiliation: Dipartimento di Matematica, Università di Tor Vergata, Viale della Ric. Scientifica, 16132 Roma, Italy
Email: cilibert@axp.mat.utovrm.it

Margarida Mendes Lopes
Affiliation: Dipartimento di Matemática, Faculdade de Ciencias de Lisboa, R. Ernesto de Vasconcelos, 1700 Lisboa, Portugal
Email: mmlopes@ptmat.lmc.fc.ul.pt

DOI: https://doi.org/10.1090/S0002-9947-98-01948-5
Received by editor(s): February 22, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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