Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Towards a Halphen theory of
linear series on curves

Authors: L. Chiantini and C. Ciliberto
Journal: Trans. Amer. Math. Soc. 351 (1999), 2197-2212
MSC (1991): Primary 14H50
Published electronically: March 1, 1999
MathSciNet review: 1422598
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A linear series $g^{N}_{\delta }$ on a curve $C\subset \mathbf{P}^{3}$ is primary when it does not contain the series cut by planes. For such series, we provide a lower bound for the degree $\delta $, in terms of deg($C$), g($C$) and of the number $s=\min \{i:h^{0}\mathcal{I}_{C}(i)\neq 0\}$. Examples show that the bound is sharp. Extensions to the case of general linear series and to the case of curves in higher projective spaces are considered.

References [Enhancements On Off] (What's this?)

  • [A] Accola R., On Castelnuovo's inequalities for algebraic curves I, Transactions of Amer. Math. Soc. 251 (1979), 357-373. MR 80i:14008
  • [B] Basili B., Indice de specialité des groupes des points de l'espace, Bull. Soc. Math. France 124 (1996), 61-95. MR 97f:14048
  • [CCD1] Chiantini L., Ciliberto C., Di Gennaro V., The genus of projective curves, Duke Math. J. 70 (1993), 229-245. MR 94b:14027
  • [CCD2] Chiantini L., Ciliberto C., Di Gennaro V., The genus of curves in $\mathbf{P}^{4}$ satisfying certain flag conditions, Manuscr. Math. 88 (1995), 119-134. MR 96h:14044
  • [CCD3] Chiantini L., Ciliberto C., Di Gennaro V., On the genus of projective curves satisfying certain flag conditions, Boll. UMI (7) 10-B (1996), 701-732. MR 98g:14028
  • [C1] Ciliberto C., Alcune applicazioni di un classico procedimento di Castelnuovo, Sem. di Geometria 1982-83 - Univ. Bologna pages 17-43.MR 86j:14027
  • [CGO] Ciliberto C., Geramita A., Orecchia F., Remarks on a theorem of Hilbert-Burch, Boll.UMI (7)2-B (1988), 463-483. MR 90a:13037
  • [CL] Ciliberto C., Lazarsfeld R., On the uniqueness of certain linear series on the curves of some class, Springer Lecture Notes 1092 (1984), 198-213. MR 86b:14015
  • [CK] Coppens M., Kato, Non-trivial linear systems on singular plane curves., Mathematishce Nachr. 166 (1994), 71-82. MR 95k:14006
  • [Ha] Halphen G., Mémoire sur la classification des courbes gauches algébriques, J. Ec. Polytech. 52 (1882), 1-200. MR
  • [H1] Harris J., The genus of space curves, Math. Ann. 249 (1980), 191-204. MR 81i:14022
  • [H2] Harris J. (with the collaboration of D. Eisenbud), Curves in projective space, Montreal University Press (1982). MR 84g:14024
  • [L] Lazarsfeld R. (with the assistance of G.F. del Busto), Lectures on linear series, IAS/Park City Math. Series 3 (1993), Amer. Math. Soc., Providence, RI, 1997, pp. 161-219. MR 98h:14008
  • [M] Mezzetti E., Differential-geometric methods for the lifting problem and linear systems on plane curves, Journal of Algebraic Geometry 3 (1994), 375-398. MR 95d:14049
  • [P] Paoletti R., Seshadri positive curves in a smooth threefold, projective $3$-fold, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Pend. Lincei (9) Mat. Appl. 6 (1995), 259-274. MR 97d:14049

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14H50

Retrieve articles in all journals with MSC (1991): 14H50

Additional Information

L. Chiantini
Affiliation: Università di Siena, Dipartimento di Matematica, Via del Capitano 15, 53100 Siena, Italy

C. Ciliberto
Affiliation: Università di Roma "Tor Vergata", Dipartimento di Matematica, Via della Ricerca Scientifica, 00133 Roma, Italy

Received by editor(s): February 22, 1996
Published electronically: March 1, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society