Abstract
In C6, we consider a non linear system of differential equations with four invariants: two quadrics, a cubic and a quartic. Using Enriques-Kodaira classification of algebraic surfaces, we show that the affine surface obtained by setting these invariants equal to constants is the affine part of an abelian surface. This affine surface is completed by gluing to it a one genus 9 curve consisting of two isomorphic genus 3 curves intersecting transversely in 4 points.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adler, M. van Mœrbeke, P.: The algebraic integrability of geodesic flow on SO(4). Invent. Math. 67, 297–331 (1982).
Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of algebraic curves, vol. I, Springer-Verlag, Grundbehren der mathematischen Wissenschaften 267, 1985.
Barth, W.; Abelian surfaces with (1,2)-polarization. Conf. on alg. Geom. Sendai (1985). Adv. Studies in Pure Math. 10, 41–84 (1987).
Griffiths, P. Harris, J.: Principles of algebraic geometry, New York, Wiley-Interscience, 1978.
Haine, L.: Geodesic flow on SO(4) and abelian surfaces. Math. Ann. 263, 435–472 (1983).
Hille, E.: Ordinary differential equations in the complex domain. Wiley, New York, 1976.
Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Math. 79,I: 109–203, II: 205-326 (1964).
Iitaka, S.: Algebraic geometry, An introduction to birational geometry of algebraic varieties, Springer-Verlag, Berlin Heidelberg New York, 1981.
Kowalewski, S.: Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12, 177–232 (1889).
Lesfari, A.: Une approche systématique à la résolution du corps solide de Kowalewski. C.R. Acad. Sc. Paris, t. 302, série I, 347–350 (1986).
Lesfari, A.: Une approche systèematique à la rèesolution des systèmes intèegrables. Proc. de la \(2{e\over}\) èecole de gèeomèetrie-analyse (15–19 juin 1987), p. 83, EHTP, Casablanca, Maroc (Confèerences organisèees en l’honneur du Pr. A. Lichnerowicz).
Lesfari, A.: Abelian surfaces and Kowalewski’s top. Ann. scient. Ec. Norm. Sup., \(4{e\over}\) sèrie, t.21, 193–223 (1988).
Lesfari, A.: Systèmes Hamiltoniens algébriquement complètement intégrables. Preprint, Dpt. Maths. Fac. Sc. El Jadida, Maroc, 1-38 (1990).
Lesfari, A. Eléments d’analyse, Sochepress-Université, Casablanca, Maroc, 1991.
Lesfari, A.: Intersection de trois quadriques et une quartique dans CP6, à paraître.
Moser, J.: Geometry of quadrics and spectral theory. Lecture delivred at the symposium in honor of S.S. Chern, Berkeley, 1979. Springer, Berlin Heidelberg New York, 1980.
Mumford, D.: Abelian varieties. Oxford university press, Bombay-Oxford, 1974.
Mumford, D.: Algebraic geometry I, complex projective varieties, Springer-Verlag, Berlin Heidelberg New York, 1976.
Mumford, M.: Lectures on curves on an algebraic surface. Annals of Math. Studies, 59, Princeton Univ. Press, Princeton, N.J., 1966.
Mumford, M.: Tata lectures on theta II, Progress in Math., Birkhaüser-Verlag, Basel, Boston, 1982
Serre, J.P.: Groupes algébriques et corps de classes, Hermann, Paris, 1959.
Weil, A.: Courbes algébriques et variétés abéliennes, Hermann, Paris, 1971.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lesfari, A. On Affine Surface That can be Completed by A Smooth Curve. Results. Math. 35, 107–118 (1999). https://doi.org/10.1007/BF03322026
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322026