Seshadri positive curves in a smooth projective 3 -fold

Roberto Paoletti

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1995)

  • Volume: 6, Issue: 4, page 259-274
  • ISSN: 1120-6330

Abstract

top
A notion of positivity, called Seshadri ampleness, is introduced for a smooth curve C in a polarized smooth projective 3 -fold X , A , whose motivation stems from some recent results concerning the gonality of space curves and the behaviour of stable bundles on P 3 under restriction to C . This condition is stronger than the normality of the normal bundle and more general than C being defined by a regular section of an ample rank- 2 vector bundle. We then explore some of the properties of Seshadri-ample curves.

How to cite

top

Paoletti, Roberto. "Seshadri positive curves in a smooth projective \( 3 \)-fold." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6.4 (1995): 259-274. <http://eudml.org/doc/244313>.

@article{Paoletti1995,
abstract = {A notion of positivity, called Seshadri ampleness, is introduced for a smooth curve \( C \) in a polarized smooth projective \( 3 \)-fold \( (X,A) \), whose motivation stems from some recent results concerning the gonality of space curves and the behaviour of stable bundles on \( \mathbb\{P\}^\{3\} \) under restriction to \( C \). This condition is stronger than the normality of the normal bundle and more general than \( C \) being defined by a regular section of an ample rank-\( 2 \) vector bundle. We then explore some of the properties of Seshadri-ample curves.},
author = {Paoletti, Roberto},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Seshadri constant; Ampleness; Bigness; Normal bundle; Cohomogical dimension; curves in a projective threefold; Seshadri-big curve; Seshadri-ample curve},
language = {eng},
month = {12},
number = {4},
pages = {259-274},
publisher = {Accademia Nazionale dei Lincei},
title = {Seshadri positive curves in a smooth projective \( 3 \)-fold},
url = {http://eudml.org/doc/244313},
volume = {6},
year = {1995},
}

TY - JOUR
AU - Paoletti, Roberto
TI - Seshadri positive curves in a smooth projective \( 3 \)-fold
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1995/12//
PB - Accademia Nazionale dei Lincei
VL - 6
IS - 4
SP - 259
EP - 274
AB - A notion of positivity, called Seshadri ampleness, is introduced for a smooth curve \( C \) in a polarized smooth projective \( 3 \)-fold \( (X,A) \), whose motivation stems from some recent results concerning the gonality of space curves and the behaviour of stable bundles on \( \mathbb{P}^{3} \) under restriction to \( C \). This condition is stronger than the normality of the normal bundle and more general than \( C \) being defined by a regular section of an ample rank-\( 2 \) vector bundle. We then explore some of the properties of Seshadri-ample curves.
LA - eng
KW - Seshadri constant; Ampleness; Bigness; Normal bundle; Cohomogical dimension; curves in a projective threefold; Seshadri-big curve; Seshadri-ample curve
UR - http://eudml.org/doc/244313
ER -

References

top
  1. ARBARELLO, E. - CORNALBA, M. - GRIFFITHS, P. - HARRIS, J., The Geometry of Algebraic Curves. Vol. I, Springer-Verlag, 1985. Zbl1235.14002MR770932
  2. BOGOMOLOV, F., Unstable vector bundles and curves on surfaces. Proc. Int. Congr. Mathem., Helsinki1978, 517-524. Zbl0485.14004MR562649
  3. EISENBUD, D. - VAN DE VEN, A., On the normal bundle of smooth rational space curves. Math. Ann., 256, 1981, 453-463. Zbl0443.14015MR628227DOI10.1007/BF01450541
  4. FUJITA, T., On the hyperplane principle of Lefshetz. J. Math. Soc. Japan, 32, 1980, 153-165. Zbl0414.14007MR554521DOI10.2969/jmsj/03210153
  5. FULTON, W., Intersection Theory. Springer-Verlag, 1984. Zbl0885.14002MR732620
  6. FULTON, W. - LAZARSFELD, R., Positivity and excess intersection. in : Enumerative and Classical Algebraic Geometry. Prog, in Math., 24, (Nice 1981) Birkhäuser, 1982, 97-105. Zbl0501.14003MR685765
  7. FULTON, W. - LAZARSFELD, R., On the connectedness of degeneracy loci and special divisors. Acta Math., 146, 1981, 271-283. Zbl0469.14018MR611386DOI10.1007/BF02392466
  8. FULTON, W. - LAZARSFELD, R., Positive polynomials for ample vector bundles. Ann of Math., 118, 1983, 35-60. Zbl0537.14009MR707160DOI10.2307/2006953
  9. HARTSHORNE, R., Ample Subvarieties of Algebraic Varieties. LNM, 156, Springer-Verlag, 1970. Zbl0208.48901MR282977
  10. KAWAMATA, Y. - MATSUDA, K. - MATSUHE, K., Introduction to the minimal model problem. In: T. ODA (ed.), Algebraic Geometry. Sendai 1985, Adv. St. in Pure Math., vol. 10, North-Holland, 1987, 283-360. Zbl0672.14006MR946243
  11. LAZARSFELD, R., Some applications of the theory of ample vector bundles. In: S. GRECO - R. STRANO (eds.), Complete Intersections. Arcireale 1983, LNM1092, Springer-Verlag, 1984, 29-61. Zbl0547.14009MR775876DOI10.1007/BFb0099356
  12. LAZARSFELD, R., A sampling of vector bundle techniques in the study of linear series. In: M. CORNALBA et al. (eds.), Proceedings of the Intern. Centre Theor. Phys. College on Riemann Surfaces (Trieste 1987). World Scientific Press, 1989, 500-559. Zbl0800.14003MR1082360
  13. LOPEZ, A., Noether-Lefshetz theory and the Picard group of projective surface. PhD Thesis, Brown University, 1988. Zbl0736.14012
  14. PAOLETTI, R., Free pencils on divisors. Mathematische Annalen, to appear. Zbl0835.14005MR1348358DOI10.1007/BF01460982
  15. PAOLETTI, R., Seshadri constants, gonality of space curves and restriction of stable bundles. J. Diff. Geom., 40, 1972, 475-504. Zbl0811.14034MR1305979
  16. PARANJAPE, K. - RAMANAN, S., On the canonical ring of a curve. In: Algebraic Geometry and Commutative Algebra, in Honor of Masayoshi Nagata. Kinokuniya, Tokio 1988, vol. II, 503-516. Zbl0699.14041MR977775
  17. SERRANO, F., Extension of morphisms defined on a divisor. Math. Ann., 277, 1987, 395-413. Zbl0595.14005MR891582DOI10.1007/BF01458322
  18. SHIFFMAN, B. - SOMMESE, A. J., Vanishing theorems on complex manifolds. Progr. Math., vol. 56, Birkhäuser, Boston1985. Zbl0578.32055MR782484
  19. SOMMESE, A. J., On manifolds that cannot be ample divisors. Math. Ann., 221, 1987, 55-72. Zbl0306.14006MR404703
  20. SPEISER, R. D., Cohomological dimension of abelian varieties. Thesis, Cornell University, 1970. Zbl0271.14009MR2619546

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.