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 with degree d and genus g has a stable normal bundle .
@article{bwmeta1.element.bwnjournal-article-apmv72z1p33bwm,
author = {Ballico, Edoardo and Ramella, Luciana},
title = {On the existence of curves in $P^n$ with stable normal bundle},
journal = {Annales Polonici Mathematici},
volume = {72},
year = {1999},
pages = {33-42},
zbl = {0978.14025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv72z1p33bwm}
}
Ballico, Edoardo; Ramella, Luciana. On the existence of curves in $ℙ^n$ with stable normal bundle. Annales Polonici Mathematici, Tome 72 (1999) pp. 33-42. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv72z1p33bwm/
[000] [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. | Zbl 0084.17305
[001] [2] E. Ballico, The subbundles of decomposable vector bundles over an elliptic curve, Collect. Math. 49 (1998), 185-189. | Zbl 0928.14022
[002] [3] E. Ballico and G. Hein, On the stability of the restriction of to projective curves, Arch. Math. (Basel), to appear. | Zbl 0921.14018
[003] [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.
[004] [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. | Zbl 0768.14012
[005] [6] D. Eisenbud and J. Harris, Finite projective schemes in linearly general position, J. Algebraic Geom. 1 (1992), 15-30. | Zbl 0804.14002
[006] [7] D. Eisenbud and A. Van de Ven, On the normal bundles of smooth rational space curves, Math. Ann. 256 (1981), 453-463. | Zbl 0443.14015
[007] [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. | Zbl 0492.14016
[008] [9] G. Ellingsrud et A. Hirschowitz, Sur le fibré normal des courbes gauches, C. R. Acad. Sci. Paris 299 (1984), 245-248. | Zbl 0572.14007
[009] [10] F. Ghione and G. Sacchiero, Normal bundle of rational curves in ℙ³, Manuscripta Math. 33 (1980), 111-128. | Zbl 0496.14021
[010] [11] R. Hartshorne, Algebraic Geometry, Springer, New York, 1977.
[011] [12] R. Hartshorne and A. Hirschowitz, Smoothing algebraic space curves, in: Algebraic Geometry (Sitges, 1983), Lecture Notes in Math. 1124, Springer, 1984, 98-131.
[012] [13] G. Hein and H. Kurke, Restricted tangent bundle of space curves, in: Israel Math. Conf. Proc. 9, 1996, 283-294. | Zbl 0859.14011
[013] [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).
[014] [15] G. Sacchiero, Normal bundles of rational curves in projective space, Ann. Univ. Ferrara Sez. VII 26 (1980), 33-40.
[015] [16] C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques (rédigé par J. M. Drézet), Astérisque 96 (1982).