Tohoku Mathematical Journal

Local splitting families of hyperelliptic pencils, I

Tatsuya Arakawa and Tadashi Ashikaga

Full-text: Open access

Abstract

We construct local splitting families of hyperelliptic pencils so that the original complicated degenerate fiber decomposes into several simple degenerate fibers. In some sense, our trial is a generalization to hyperelliptic curves of arbitrary genus of Moishezon's construction for families of elliptic curves. Moreover, we study certain invariants of degenerate fiber germs.

Article information

Source
Tohoku Math. J. (2), Volume 53, Number 3 (2001), 369-394.

Dates
First available in Project Euclid: 3 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1178207417

Digital Object Identifier
doi:10.2748/tmj/1178207417

Mathematical Reviews number (MathSciNet)
MR2002h:14041

Zentralblatt MATH identifier
1081.14508

Subjects
Primary: 14H10: Families, moduli (algebraic)

Citation

Arakawa, Tatsuya; Ashikaga, Tadashi. Local splitting families of hyperelliptic pencils, I. Tohoku Math. J. (2) 53 (2001), no. 3, 369--394. doi:10.2748/tmj/1178207417. https://projecteuclid.org/euclid.tmj/1178207417


Export citation

References

  • [Ac] N. A'CAMPO, Le groupe de monodromie du deploiment des singularity isolees de courbes planes I, Math. Ann. 213 (1975), 1-32.
  • [AA] T. ARAKAWA AND T. ASHIKAGA, Local splitting families of hyperelliptic pencils II, preprint
  • [Ar] M. ARTIN, Algebraic approximation of structures over complete local rings, Inst. Hautes Etudes Sci. Publ Math. 36 (1969), 23-58.
  • [As] T. ASHIKAGA, Surface singularities on cyclic coverings and an inequality for the signature, J. Math. Soc Japan 51 (1999), 485-521.
  • [Atl] M. F. ATIYAH, On analytic surfaces with double points, Proc. Roy. Soc. London Ser. A 247 (1958), 237-244.
  • [At2] M. F. ATIYAH, The signature of fiber bundles, Global Analysis (ed. D. C. Spencer and S. Iyanaga), 73-84, Univ. Tokyo Press and Princeton Univ. Press, Tokyo, Princeton, 1969.
  • [B] E. BRIESKORN, Uber die Anflosung gewisser Singularitaten von holomorphen Abbildungen, Math. Ann 166 (1966), 76-102.
  • [C] Z. CHEN, On the geography of surfaces, Math. Ann. 277 (1987), 141-164
  • [E] H. ENDO, Meyer's signature cocycle and hyperelliptic fibrations, Math. Ann. 316 (2000), 237-257
  • [FM] R. FRIEDMAN AND J. MORGAN, Smooth four-manifolds and complex surfaces, Ergeb. Math. Grenzgeb (3) 27, Springer-Verlag, Berlin, 1994.
  • [FN] A. FUJIKI AND S. NANANO, Supplement to "On the inverse of monoidal transformation", Publ. Res. Inst Math. Sci. 7 (1971/72), 637-644.
  • [G] S. M. GUSEIN-ZADE, Intersection matrices for certain singularities of functions of two variables, Funk sional Anal, i Prilozhen. 8 (1974), 11-15.
  • [Hil] F. HIRZEBRUCH, Topological Methods in Algebraic Geometry, Grundlehren Math. Wiss. 131, Springer Verlag, New York Inc., New York, 1966.
  • [Hi2] F. HIRZEBRUCH, The signature of ramified coverings, Global Analysis (ed. D. C. Spencer and S. Iyanaga), 253-265, Univ. Tokyo Press and Princeton Univ. Press, Tokyo, Princeton, 1969.
  • [Hoi] E. HORIKAWA, On deformations of quintic surfaces, Invent. Math. 31 (1975), 43-85
  • [Ho2] E. HORIKAWA, On algebraic surfaces with pencils of curves of genus 2, Complex analysis and algebrai geometry, a collection of papers dedicated to K. Kodaira (ed. W. L. Baily, Jr. and T. Shioda), 79-90, Iwanami Shoten and Cambridge Univ. Press, Tokyo, Cambridge-New York, 1977.
  • [Ho3] E. HORIKAWA, Algebraic surfaces of general type with small cj, V, J. Fac. Sci. Univ. Tokyo Sect. IA Math 28 (1981), 745-755.
  • [Ho4] E. HORIKAWA, Local deformation of pencils of curves of genus two, Proc. Japan Acad. Ser. A Math. Sci 64(1988), 241-244.
  • [Kodl] K. KODAIRA, On compact complex analytic surfaces II, Ann. of Math. 77 (1963), 563-626
  • [Kod2] K. KODAIRA, A certain type of irregular algebraic surfaces, J. Anal. Math. 19 (1967), 207-215
  • [Kon] K. KONNO, Non-hyperelliptic fibrations of small genus and certain irregular canonical surfaces, Ann Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), 575-595.
  • [Mai] Y. MATSUMOTO, Good torus fibrations, Contemp. Math. 35(1984), 375-397
  • [Ma2] Y. MATSUMOTO, Torus fibrations over the 2-sphere with the simplest singular fibers, J. Math. Soc. Japa 37 (1985), 605-636.
  • [Ma3] Y. MATSUMOTO, Diffeomorphism type of elliptic surfaces, Topology 25 (1986), 549-563
  • [Ma4] Y. MATSUMOTO, Lefschetz fibrations of genus two--a topological approach, Proc. of the 37th Taniguch Symposium on Topology and Teichmiiller spaces (Katinkulta, 1995), 123-148, World Sci. Publishing, River Edge, NJ, 1996.
  • [Ma5] Y. MATSUMOTO, Splitting ofcertain singular fibers ofgenus 2, Preprint
  • [MM] Y. MATSUMOTO AND J. M. MONTESINOS-AMILIBIA, Pseudo-periodic maps and degeneration of Riemann surfaces I, II, Preprints, Univ. of Tokyo and Univ. Complutense de Madrid, 1991/1992.
  • [Men] M. MENDES-LOPES, The relative canonical algebra for genus 3fibrations, thesis, Warwick Univ., 1988
  • [Mey] W. MEYER, Die Signatur von Flachenbndeln, Math. Ann. 201 (1973), 239-264
  • [Moi] B. MOISHEZON, Complex surfaces and connected sums ofcomplex projective planes, Lecture Notes i Math. 603, Springer-Verlag, Berlin-New York, 1977.
  • [Mor] T. MORIFUJI, Meyer's function, ^-invariants and the signature cocycle, thesis, Univ. of Tokyo, 1998
  • [Na] N. NAKAYAMA, Global structure ofan elliptic fibration, preprint (RIMS-1322), 2001
  • [NU] Y. NAMIKAWA AND K. UENO, The complete classification of fibers inpencils ofcurves ofgenus two, Manuscripta Math. 9(1973), 143-186.
  • [PI] U. PERSSON, Double coverings and surfaces ofgeneral type, Algebraic Geometry Proceedings 1977 (ed L. D. Olson), 168-195, Lecture Notes in Math. 687, Springer-Verlag, Berlin, 1978.
  • [P2] U. PERSSON, On Chern invariants ofsurfaces ofgeneral type, Compositio Math.43 (1981), 3-58
  • [P3] U. PERSSON, Genus two fibration revisited, Complex algebraic varieties (Bayreuth, 1990), 133-144, Lec ture Notes in Math. 1507, Springer-Verlag, Berlin, 1992.
  • [Re] M. REID, Problems on pencils ofsmall genus, Preprint 1990
  • [Uel] M. UE, Splitting singular fibers in good torus fibrations, J. Fac.Sci. Univ. Tokyo Sect. IA Math. 32 (1985), 165-204.
  • [Ue2] M. UE, On the diffeomorphism type of elliptic surfaces with multiple fibers, Invent. Math. 84(1986), 633-643.
  • [Uen] K. UENO, Discriminants ofcurves ofgenus 2and arithmetic surfaces, in Algebraic geometry andcommu tative algebra inhonor of M. Nagata (ed. H. Hijikata, H. Hironaka, M. Maruyama, H. Matsumura, M. Miyanishi, T. Oda and K. Ueno), 749-770, Kinokuniya, Tokyo, 1988.
  • [XI] G. XIAO, Surfaces fibrees en courbes de genre deux, Lecture Notes in Math. 1137, Springer-Verlag, Berlin New York, 1985.
  • [X2] G. XIAO, Fibered algebraic surfaces with low slope, Math.Ann. 276 (1987), 449^66
  • [X3] G. XIAO, The fibrations of algebraic surfaces (in Chinese), Shanghai Scientific echnical Publishers, 1992.