On the existence of curves in ℙ n with stable normal bundle

Edoardo Ballico; Luciana Ramella

Annales Polonici Mathematici (1999)

  • Volume: 72, Issue: 1, page 33-42
  • ISSN: 0066-2216

Abstract

top
We prove that for integers n,d,g such that n ≥ 4, g ≥ 2n and d ≥ 2g + 3n + 1, the general (smooth) curve C in ℙ n with degree d and genus g has a stable normal bundle N C .

How to cite

top

Ballico, Edoardo, and Ramella, Luciana. "On the existence of curves in $ℙ^n$ with stable normal bundle." Annales Polonici Mathematici 72.1 (1999): 33-42. <http://eudml.org/doc/262558>.

@article{Ballico1999,
abstract = {We prove that for integers n,d,g such that n ≥ 4, g ≥ 2n and d ≥ 2g + 3n + 1, the general (smooth) curve C in $ℙ^n$ with degree d and genus g has a stable normal bundle $N_C$.},
author = {Ballico, Edoardo, Ramella, Luciana},
journal = {Annales Polonici Mathematici},
keywords = {elementary transformations of bundles on smooth curves; bundles on nodal curves; normal bundle; nodal curves; bundles on smooth curves; stable normal bundle},
language = {eng},
number = {1},
pages = {33-42},
title = {On the existence of curves in $ℙ^n$ with stable normal bundle},
url = {http://eudml.org/doc/262558},
volume = {72},
year = {1999},
}

TY - JOUR
AU - Ballico, Edoardo
AU - Ramella, Luciana
TI - On the existence of curves in $ℙ^n$ with stable normal bundle
JO - Annales Polonici Mathematici
PY - 1999
VL - 72
IS - 1
SP - 33
EP - 42
AB - We prove that for integers n,d,g such that n ≥ 4, g ≥ 2n and d ≥ 2g + 3n + 1, the general (smooth) curve C in $ℙ^n$ with degree d and genus g has a stable normal bundle $N_C$.
LA - eng
KW - elementary transformations of bundles on smooth curves; bundles on nodal curves; normal bundle; nodal curves; bundles on smooth curves; stable normal bundle
UR - http://eudml.org/doc/262558
ER -

References

top
  1. [1] M. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414-452; reprinted in: M. Atiyah, Collected Works, Vol. 1, Oxford Sci. Publ., Oxford, 1988, 105-143. Zbl0084.17305
  2. [2] E. Ballico, The subbundles of decomposable vector bundles over an elliptic curve, Collect. Math. 49 (1998), 185-189. Zbl0928.14022
  3. [3] E. Ballico and G. Hein, On the stability of the restriction of T ℙ n to projective curves, Arch. Math. (Basel), to appear. Zbl0921.14018
  4. [4] A. Bruguières, Filtration de Harder-Narasimhan et stratification de Shatz, in: Modules des fibrés stables sur les courbes algébriques, Progr. Math. 54, Birkhäuser, 1985, 81-104. 
  5. [5] L. Ein and R. Lazarsfeld, Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves, in: Complex Projective Geometry (Trieste 1989 / Bergen 1989), G. Ellingsrud, C. Peskine, G. Sacchiero and S. A. Stromme (eds.), London Math. Soc. Lecture Note Ser. 179, Cambridge Univ. Press, Cambridge, 1992, 149-156. Zbl0768.14012
  6. [6] D. Eisenbud and J. Harris, Finite projective schemes in linearly general position, J. Algebraic Geom. 1 (1992), 15-30. Zbl0804.14002
  7. [7] D. Eisenbud and A. Van de Ven, On the normal bundles of smooth rational space curves, Math. Ann. 256 (1981), 453-463. Zbl0443.14015
  8. [8] D. Eisenbud and A. Van de Ven, On the variety of smooth rational space curves with given degree and normal bundle, Invent. Math. 67 (1982), 89-100. Zbl0492.14016
  9. [9] G. Ellingsrud et A. Hirschowitz, Sur le fibré normal des courbes gauches, C. R. Acad. Sci. Paris 299 (1984), 245-248. Zbl0572.14007
  10. [10] F. Ghione and G. Sacchiero, Normal bundle of rational curves in ℙ³, Manuscripta Math. 33 (1980), 111-128. Zbl0496.14021
  11. [11] R. Hartshorne, Algebraic Geometry, Springer, New York, 1977. 
  12. [12] R. Hartshorne and A. Hirschowitz, Smoothing algebraic space curves, in: Algebraic Geometry (Sitges, 1983), Lecture Notes in Math. 1124, Springer, 1984, 98-131. 
  13. [13] G. Hein and H. Kurke, Restricted tangent bundle of space curves, in: Israel Math. Conf. Proc. 9, 1996, 283-294. Zbl0859.14011
  14. [14] L. Ramella, Sur les schémas définissant les courbes rationnelles lisses de ℙ³ ayant fibré normal et fibré tangent restreint fixés, Mém. Soc. Math. France 54 (1993). 
  15. [15] G. Sacchiero, Normal bundles of rational curves in projective space, Ann. Univ. Ferrara Sez. VII 26 (1980), 33-40. 
  16. [16] C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques (rédigé par J. M. Drézet), Astérisque 96 (1982). 

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.