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Bicanonical pencil of a determinantal Barlow surface

Author: Yongnam Lee
Journal: Trans. Amer. Math. Soc. 353 (2001), 893-905
MSC (2000): Primary 14J10, 14J29
Published electronically: November 8, 2000
MathSciNet review: 1707700
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Abstract | References | Similar Articles | Additional Information


In this paper, we study the bicanonical pencil of a Godeaux surface and of a determinantal Barlow surface. This study gives a simple proof for the unobstructedness of deformations of a determinantal Barlow surface. Then we compute the number of hyperelliptic curves in the bicanonical pencil of a determinantal Barlow surface via classical Prym theory.

References [Enhancements On Off] (What's this?)

  • [AC] Enrico Arbarello and Maurizio Cornalba, The Picard groups of the moduli spaces of curves, Topology 26 (1987), no. 2, 153–171. MR 895568, 10.1016/0040-9383(87)90056-5
  • [Ba] Rebecca Barlow, A simply connected surface of general type with 𝑝_{𝑔}=0, Invent. Math. 79 (1985), no. 2, 293–301. MR 778128, 10.1007/BF01388974
  • [B1] Arnaud Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309–391 (French). MR 0472843
  • [B2] Arnaud Beauville, L’application canonique pour les surfaces de type général, Invent. Math. 55 (1979), no. 2, 121–140 (French). MR 553705, 10.1007/BF01390086
  • [B3] 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
  • [C1] F. Catanese, Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications, Invent. Math. 63 (1981), no. 3, 433–465. MR 620679, 10.1007/BF01389064
  • [C2] F. Catanese, Pluricanonical mapping of surfaces with $K^{2}=1,2$, $q=p_{g}=0$, C.I.M.E. 1977 Algebraic Surfaces, Liguori, Napoli (1981), 249-266.
  • [CL] Fabrizio Catanese and Claude LeBrun, On the scalar curvature of Einstein manifolds, Math. Res. Lett. 4 (1997), no. 6, 843–854. MR 1492124, 10.4310/MRL.1997.v4.n6.a5
  • [CP] F. Catanese and R. Pignatelli, On simply connected Godeaux surfaces, Preprint (1999).
  • [Fu] Takao Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), no. 4, 779–794. MR 513085, 10.2969/jmsj/03040779
  • [HM] Joe Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23–88. With an appendix by William Fulton. MR 664324, 10.1007/BF01393371
  • [Ko] János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180
  • [L1] Y. Lee, Degeneration of numerical Godeaux surfaces, Ph.D. Thesis, University of Utah (1997).
  • [L2] Y. Lee, A compactification of a family of determinantal Godeaux surfaces, Trans. Amer. Math. Soc. 352 (2000), 5013-5023. CMP 98:13
  • [Mi] Yoichi Miyaoka, Tricanonical maps of numerical Godeaux surfaces, Invent. Math. 34 (1976), no. 2, 99–111. MR 0409481
  • [Re] Miles Reid, Surfaces with 𝑝_{𝑔}=0, 𝐾²=1, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1978), no. 1, 75–92. MR 494596
  • [Te] M. Texidor, The divisor of curves with a vanishing theta-null, Compositio Math. 66 (1988), 15-22.

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Additional Information

Yongnam Lee
Affiliation: Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, Korea
Address at time of publication: Department of Mathematics, Sogang University, Sinsu-dong, Mapo-gu, Seoul 121-742, Korea

Received by editor(s): October 15, 1997
Received by editor(s) in revised form: June 3, 1999
Published electronically: November 8, 2000
Additional Notes: The author would like to express his appreciation to professor Herb Clemens for valuable suggestions that made this work possible. Also the author would like to thank Professor Fabrizio Catanese for the use of his approach and the results in his recent preprint (written with Pignatelli) to solve the problem of hyperelliptic curves. Finally, the author would like to thank the referee for many interesting suggestions and corrections. Most of this work is the part of a Ph.D. thesis submitted to the University of Utah in 1997, and is partially supported by the Korea Institute for Advanced Study.
Article copyright: © Copyright 2000 American Mathematical Society