Abstract
Let X be an irreducible algebraic curve of genus g smooth and proper over an algebraically closed field k, ℰ a locally free sheaf of rank 2 over X, F=P(ℰ) the projective bundle associated to ℰ and ρ:F→X the canonical projection. Aunisecant curve on F is a curve (effective divisor) C on F such that the intersection number (C,ρ−1(x))=1, x ∈ X. Notice that a section of F over X or alternatively a sub-line bundle of ℰ means simply an irreducible unisecant curve. We give here some results on unisecant curves on F. In particular we are able to prove C.Segre's result regarding his “general surfaces” [8]. A more ample account including all the details will appear later.
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References
Grothendieck, A.: Technique de descente et théorèmes d'existence en géométrie algébrique. V. Les schémas de Picard: Théorémes d'existence. Séminaire Bourbaki. 14e année, no. 232, (1961/62)
Hartshorne, R.: Ample vector bundles. Publ. Math. IHES29, 63–94 (1966)
Kleiman, S. and Laksov, D.: Another proof of the existence of special divisors. Acta Math.132, 163–176 (1974)
Mattuck, A.: On symmetric product of curves. Proc. Am. Math. Soc.13, 82–87 (1962)
Mumford, D.: Lectures on curves on an algebraic surface. Princeton Univ. Press 1966
Nagata, M.: On ruled surfaces. Séminaire de Mathématiques Supérieures, Université de Montréal46, Second part, 1970
Narasimhan M.S. and Ramanan, S.: Moduli of vector bundles on a compact Riemann surface. Annals of Math.89, 14–51 (1960)
Segre, C.: Recherches générales sur les courbes et les surfaces réglées algébriques. II. Math. Annalen34, 1–25 (1889)
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Ghione, F., Lascu, A.T. Unisecant curves on a general ruled surface. Manuscripta Math 26, 169–177 (1978). https://doi.org/10.1007/BF01167972
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DOI: https://doi.org/10.1007/BF01167972