Hostname: page-component-7b9c58cd5d-9klzr Total loading time: 0 Render date: 2025-03-15T00:50:11.504Z Has data issue: false hasContentIssue false
  • English
  • Français

The genus of curves on the three dimensional quadric

Published online by Cambridge University Press:  22 January 2016

Mark Andrea A. De Cataldo
Mark Andrea A. De Cataldo*
Affiliation:
Department of Mathematics, Washington University in St. Louis Campus, Box 1146 Saint Louis 63130 (MO), U. S. A.mde@math.wustl.edu
Rights & Permissions [Opens in a new window]

Abstract

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.

By means of an ad hoc modification of the so-called “Castelnuovo-Harris analysis” we derive an upper bound for the genus of integral curves on the three dimensional nonsingular quadric which lie on an integral surface of degree 2/c, as a function of k and the degree d of the curve. In order to obtain this we revisit the Uniform Position Principle to make its use computation-free. The curves which achieve this bound can be conveniently characterized.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 147 , September 1997 , pp. 193 - 211
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1997

References

[A-C-G-H] Arbarello, E., Cornalba, M., Griffiths, P. A., Harris, J., Geometry of Algebraic Curves I, Springer, 1985.Google Scholar
[A-S] Arrondo, E., Sols, I., On Congruences of Lines in the Projective Space, fascicule 3., Mémoire n. 50 Supplément au Bulletin de la S. M. F., 120 (1992), Société Mathématique de France.Google Scholar
[C-C-D] Chiantini, L., Ciliberto, C., Di Gennaro, V., The genus of projective curves, Duke Math. J., 70 (1993), no. 2, 229–245.Google Scholar
[C-K-M] Clemens, H., Kollár, J., Mori, S., Higher Dimensional Complex Geometry, Astérisque, 166, Société Mathématique de France (1988).Google Scholar
[D1] Cataldo, M. A. de, Codimension two subvarieties of quadrics, Ph. D. Thesis Notre Dame (1995).Google Scholar
[D2] Cataldo, M. A. de, A fimteness theorem for codimension two nonsingular sub-varieties of quadrics, Trans. Amer. Math. Soc, 349 (1997), 2359–2370.Google Scholar
[D3] Cataldo, M. A. de, Some adjunction-theoretic properties of codimension two nonsingular subvarieties of quadrics, to appear in Can. Jour, of Math..Google Scholar
[D4] Cataldo, M. A. de, Codimension two nonsingular subvarieties of quadrics: scrolls and classification in degree d ≤ 10, preprint, alg-geom eprints 960–8021.Google Scholar
[E-P] Ellinsgrud, G., Peskine, C., Sur les surfaces lisses de P4 , Invent. Math., 95 (1989), 1–11.Google Scholar
[G-P] Gruson, L., Peskine, C., Genre des courbes de l’espace projectif in Proceedings of Tromso (Conference on Algebraic Geometry), LNM 687, Springer (1977), pp. 31–59.Google Scholar
[Ha] Hartshorne, R., Algebraic Geometry, GTM 52, Springer, 1977.Google Scholar
[JH] Harris, J., The Genus of Space Curves, Math. Ann., 249 (1980), 191–204.Google Scholar
[Mi] Migliore, J., Liaison of a union of Skew Lines in P4 , Pac. Jour. Math., 130 (1987), no. 1, 153–170.Google Scholar
[Mu] Mumford, D., Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, 59 (1966), Princeton Univ. Press.Google Scholar