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Existence of coherent systems. (English) Zbl 1161.14024

Coherent systems (vector bundle \(E\) with a subspace \(V\) of its space of sections \(H^0(E)\)) on a curve are considered. The author investigates the condition of (semi)stability of coherent system \((E,V)\) of fixed degree \(d\), number of sections \(k\) and rank \(r\), depending on a real parameter \(\alpha\). The set of these objects can be endowed with a structure of moduli space.The conditions for non-emptiness of such moduli spaces for \(k\) small were given by other authors [L. Brambila- Paz, Int. J. Math. 19, No. 7, 777–799 (2008; Zbl 1173.14023); S. B. Bradlow, O. García-Prada, V. Muñoz and P. E. Newstead, Int. J. Math. 14, No. 7, 683–733 (2003; Zbl 1057.14041)].
The present paper deals with the case \(d+r(1-g)\geq k>r\), \(g\geq r+2.\)
Relevant paper of the author: M. Teixidor i Bigas [Math. Ann. 290, No. 2, 341–348 (1991; Zbl 0719.14015)].

MSC:

14H60 Vector bundles on curves and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H51 Special divisors on curves (gonality, Brill-Noether theory)
14H52 Elliptic curves
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[1] DOI: 10.1142/S0129167X03002009 · Zbl 1057.14041 · doi:10.1142/S0129167X03002009
[2] DOI: 10.1142/S0129167X95000316 · Zbl 0861.14028 · doi:10.1142/S0129167X95000316
[3] Teixidor M., Adv. Geom. 5 pp 37–
[4] Teixidor M., Duke Math. J. 62 pp 385–
[5] DOI: 10.1007/BF01459249 · Zbl 0719.14015 · doi:10.1007/BF01459249
[6] Teixidor M., Amer. J. Math. 219 pp 477–
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