Stability of the bundles \(\Lambda^p E_L\) and the Raynaud condition. (Stabilité des fibrés \(\Lambda^p E_L\) et condition de Raynaud.) (French. English summary) Zbl 1084.14519
Summary: Let \(C\) be a smooth curve of genus \(g \geq 2\) on \(C\). Let \(L\) be a line bundle on \(C\) generated by its global sections and let \(E_{L}\) be the dual of the kernel of the evaluation map \(e_{L}\). We are studying here the relation between the stability the fact that the bundle is verifying a condition \((R)\) introduced by
M. Raynaud [Bull. Soc. Math. Fr. 110, 103–125 (1982; Zbl 0505.14011)]: we prove that \(E_{L}\) is semi-stable when \(C\) is general. We also prove that \(E_{L}\) is verifying \((R)\) when \(\deg(L) \geq 2g\) or when \(L\) is generic. Finally we prove that for each \(p\) in \(\{2,..., \text{rg}(E_{L})-2\}\), if \(\deg(L) \geq 2g+2\) then \(\Lambda^{p}E_{L}\) is not verifying \((R)\).
MSC:
| 14H60 | Vector bundles on curves and their moduli |
Citations:
Zbl 0505.14011References:
| [1] | Arbarello, E., CORNALBA, Griffiths, P., J.Harris, J. - Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, 267, Springer-Verlag, New York (1985). · Zbl 0559.14017 |
| [2] | Ballico, E. - Line bundles on projective curves : the multiplication map, Atti Sem. Mat. Fis. Univ Modena, L, p. 17-21 (2002). · Zbl 1221.14040 |
| [3] | Beauville, A. - Some stable vector bundles with reducible theta divisors, Manuscripta Math.110, p. 343-349 (2003). · Zbl 1016.14016 |
| [4] | Beauville, A. - Vector bundles on curves and generalized theta functions : recent results and open problems, Current topics in complex algebraic geometry (Berkeley, CA, 1992/93, Math. Sci. Res. Inst. Publ., 28, Cambridge Univ. Press, Cambridge, p. 17-33 (1995). · Zbl 0846.14024 |
| [5] | Ein, L. , Lazarsfeld, R. - Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves, 179, Cambridge Univ. Press, Cambridge (1992). · Zbl 0768.14012 |
| [6] | Lazarsfeld, R. - A sampling of vector bundle techniques in the study of linear series, Lectures , World scientific press, Singapore , p. 500-559 (1989). · Zbl 0800.14003 |
| [7] | Popa, M. - On the base locus of the generalized theta divisor , C. R. Acad. Sci. Paris Sér. I Math.329, no. 6, p. 507-512 (1999). · Zbl 0959.14020 |
| [8] | Raynaud, M. - Sections des fibrés vectoriels sur une courbe, Bull. Soc. math. France, 110, p. 103-125 (1982). · Zbl 0505.14011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
