Hostname: page-component-66644f4456-mv4w6 Total loading time: 0 Render date: 2025-02-13T10:50:06.638Z Has data issue: true hasContentIssue false
  • English
  • Français

On the fields of rationality for curves and for their jacobian varieties

Published online by Cambridge University Press:  22 January 2016

Tsutomu Sekiguchi
Tsutomu Sekiguchi*
Affiliation:
Department of Mathematics, Faculty of Science and Engineering Chuo University, Kasuga, Bunkyo-ku Tokyo, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Throughout the paper, a scheme means a noetherian scheme. By a curve C over a scheme S of genus g, we mean a proper and smooth S-scheme with irreducible curves of genus g as geometric fibres. In the previous paper [15], the author showed that the field of moduli for a non-hyperelliptic curve over a field coincides with the one for its canonically polarized jacobian variety, and in [16], he gave a partial result on the coincidence of the fields of rationality for a hyperelliptic curve and for its canonically polarized jacobian variety. In the present paper, we will discuss the isomorphy of the isomorphism schemes of two curves over a scheme and of their canonically polarized jacobian schemes, by using Oort-Steenbrink’s result [12].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 88 , December 1982 , pp. 197 - 212
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1982

References

[ 1 ] Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus, publ. Math., 36 (Volume dedicated to 0. Zariski), I.H.E.S. (1969), 75–109.Google Scholar
[ 2 ] Grothendieck, A., Fondements de la géométrie algébrique, Séminaire Bourbaki, 1952–62, Secrétariat Math., Paris (1962), Refered to as FGA.Google Scholar
[ 3 ] Grothendieck, A., Eléments de géométrie algébrique (with J. Dieudonne), Pub. Math. I.H.E.S., 1960–67, Refered to as EGA.Google Scholar
[ 4 ] Grothendieck, A. et al, Séminaire de géométrie algébrique 1, Lecture notes in Math. 224, Springer-Verlag, 1971, Refered to as SGA 1.Google Scholar
[ 5 ] Homma, M., Automorphisms of prime order on curves, Manuscripta Math., 33 (1980), 99–109.Google Scholar
[ 6 ] Koizumi, S., The fields of moduli for polarized abelian varieties and for curves, Nagoya Math. J., 48 (1972), 37–55.Google Scholar
[ 7 ] Laudal, O. A. and Lonsted, K., Deformations of curves I, Moduli for hyperelliptic curves, Algebraic geometry, Proceedings Tromso, Norway 1977, Lecture notes in Math. 687, Springer-Verlag, 1978.Google Scholar
[ 8 ] Matsusaka, T., On a theorem of Torelli, Amer. J. Math., 80 (1958), 784–800.Google Scholar
[ 9 ] Mumford, D., Geometric invariant theory, Ergebnisse, Springer-Verlag, 1965.Google Scholar
[10] Mumford, D., Varieties defined by quadratic equations, Questioni sulle Varieta Algebriche, Corsi dal C.I.M.E., Edizioni Cremonese, Roma 1969.Google Scholar
[11] Mumford, D., Abelian varieties, Tata Inst. Studies in Math., Oxford Univ. Press, London and New York, 1970.Google Scholar
[12] Oort, F. and Steenbrink, J., The local Torelli problem for algebraic curves, Univ. Utrecht, Dep. Math., Preprint Nr. 136, 1979.Google Scholar
[13] Oort, F. and Ueno, K., Principally polarized abelian varieties of dimension two and three are Jacobian varieties, J. Fac. Sci. Univ. Tokyo, Section IA, Math., 20 (1973), 377–381.Google Scholar
[14] Popp, H., Moduli theory and classification theory of algebraic varieties, Lecture notes in Math. 620, Springer-Verlag, 1977.Google Scholar
[15] Sekiguchi, T., The coincidence of fields of moduli for non-hyperelliptic curves and for their jacobian varieties, Nagoya Math. J., 82 (1981), 57–82.Google Scholar
[16] Sekiguchi, T., On the fields of rationality for curves and for abelian varieties, Bull. Fac. Sci. & Eng. Chuo Univ., 23 (1980) 35–41.Google Scholar
[17] Weil, A., Zum Beweis des Torellischen Satzes, Nachr. Akad. Wissensch. Göttingen, (1957), 33–53.Google Scholar