Linear pencils on real algebraic curves |
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Authors: | Marc Coppens Gerriet Martens |
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Institution: | a Katholieke Industriele Hogeschool der Kempen, Dept. IBW, Afdeling Algebra KU Leuven, Kleinhoefstraat 4, B - 2440 Geel, Belgium b Department Mathematik, Universität Erlangen-Nürnberg, Bismarckstr. 1 12, D-91054 Erlangen, Germany |
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Abstract: | Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n. |
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Keywords: | 14H51 14P05 |
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