Advertisement

Non-gaps for the genera for space curves in the Range A

  • E. BallicoEmail author
Article
  • 5 Downloads

Abstract

We consider the existence of space curves with prescribed degree d and genus g and not contained in a surface of prescribed degree \(m-1\) when g is below the maximal genus allowed by the pair (dm), mainly in the so-called Range A and in the part \(d\le m(m+1)/2\) of Range B.

Keywords

Space curve Range A Range B Normal bundle Hilbert function 

Mathematics Subject Classification

14H50 

Notes

References

  1. 1.
    Ballico, E., Ellia, Ph.: A program for space curves. In: Conference on Algebraic Varieties of Small Dimension (Turin, 1985). Rend. Sem. Mat. Univ. Politec. Torino 1986, Special Issue, pp. 25–42 (1987)Google Scholar
  2. 2.
    Ballico, E., Bolondi, G., Ellia, Ph., Mirò-Roig, R.M.: Curves of maximum genus in the range A and stick-figures. Trans. Am. Math. Soc. 349(11), 4589–4608 (1997)Google Scholar
  3. 3.
    Ballico, E., Ellia, Ph.: The maximal genus of space curves in the Range A. arXiv:1211.08807
  4. 4.
    Ballico, E., Ellia, Ph., Fontanari, C.: Maximal rank of space curves in the Range A. Eur. J. Math. 4, 778–801 (2018)Google Scholar
  5. 5.
    Dolcetti, A.: Halphen’s gaps for space curves of submaximum genus. Bull. Soc. Math. France 116(2), 157–170 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eisenbud, D., Van de Ven, A.: On the normal bundles of smooth rational space curves. Math. Ann. 256, 453–463 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ellia, Ph.: Sur les lacunes d’Halphen. In: Algebraic Curves and Projective Geometry, Proceedings of Trento 1988, Lecture Notes in Mathmatics 1389, pp. 43-65 (1989)Google Scholar
  8. 8.
    Ellia, Ph.: Sur le genre maximal des courbes gauches de degrè \(d\) non sur une surface de degrè \(s - 1\). J. Rein. Angew. Math. 413, 78–87 (1991)Google Scholar
  9. 9.
    Ellia, Ph., Strano, R.: Sections planes et majoration du genre des courbes gauches. In: Complex Projective Geometry, Proceedings of Trieste-Bergen, London Mathmatical Society. Lecture Notes Series, 179, pp. 157–174. Cambridge University Press (1992)Google Scholar
  10. 10.
    Ellingsrud, G., Hirschowitz, A.: Sur le fibré normal des courbes gauches. C. R. Acad. Sci. Paris Sér. I Math. 299(7), 245–248 (1984)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fløystad, G.: Construction of space curves with good properties. Math. Ann. 289(1), 33–54 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fløystad, G.G.: On space curves with good cohomological properties. Math. Ann. 291(3), 505–549 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gruson, L., Peskine, Ch.: Genre des courbes de l’espace projectif, Proceedings of Tromsø 1977, Lecture Notes in Mathematics, vol. 687, pp. 39–59. Springer, Berlin (1978)Google Scholar
  14. 14.
    Halphen, G.: Mèmoire sur la classification des courbes gauches algèbriques, Oeuvres completes, t III, 261–455 (1881)Google Scholar
  15. 15.
    Hartshorne, R.: On the classification of algebraic space curves. In: Vector Bundles and Differential Equations (Nice 1979), Progress in Mathematics 7, pp. 82–112. Birkhäuser, Boston (1980)Google Scholar
  16. 16.
    Hartshorne, R.: On the classification of algebraic space curves. II. In: Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), pp. 145–164, Proceedings of Symposium Pure Mathematics, 46, Part 1. American Mathematical Society, Providence, RI (1987)Google Scholar
  17. 17.
    Hartshorne, R.: Stable reflexive sheaves, III. Math. Ann. 279(3), 517–534 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hartshorne, R., Hirschowitz, A.: Smoothing algebraic space curves. In: Algebraic Geometry, Sitges 1983. Lecture Notes in Mathematics, vol. 1124, pp. 98–131. Springer, Berlin (1985)Google Scholar
  19. 19.
    Hartshorne, R., Hirschowitz, A.: Nouvelles courbes de bon genre dans l’espace projectif. Math. Ann. 280, 353–367 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hirschowitz, A.: Existence de faisceaux réflexifs de rang deux sur \({\mathbb{P}}^3\) à bonne cohomologie. Publ. Math. IHES 66, 105–137 (1987)CrossRefzbMATHGoogle Scholar
  21. 21.
    Kleppe, J.O.: On the existence of nice components in the Hilbert scheme \(H(d, g)\) of smooth connected space curves. Boll. Un. Mat. Ital. B (7) 8(2), 305–326 (1994)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kleppe, J.O.: Halphen gaps and good space curves. Boll.U.M.I. (8) 1–B, 429–450 (1998)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Larson, E.: The generality of a section of a curve. arXiv:1605.06185
  24. 24.
    Perrin, D.: Courbes passant par \(m\) points généraux de \({\mathbb{P}}^3\). Bull. Soc. Math. France, Mémoire 28/29 (1987)Google Scholar
  25. 25.
    Strano, R.: Plane sections of curves of \({\mathbb{P}}^3\) and a conjecture of Hartshorne and Hirschowitz. Rend. Sem. Mat. Univ. Politec. Torino 48, 511–527 (1990)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Strano, R.: On the genus of a maximal rank curve in \({\mathbb{P}}^3\). J. Algebraic Geom. 3, 435–447 (1994)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Vogt, I.: Interpolation for Brill–Noether space curves. Manuscr. Math. 156, 137–147 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Walter, C.H.: The cohomology of the normal bundles of space curves, I. Preprint, Rutgers University (1989)Google Scholar
  29. 29.
    Walter, C.H.: Curves on surfaces with a multiple line. J. Reine Angew. Math. 412, 48–62 (1990)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Walter, C.H.: Curves in \({\mathbb{P}}^3\) with the expected monad. J. Algebraic Geom. 4(2), 301–330 (1995)MathSciNetGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TrentoPovoItaly

Personalised recommendations