Journal of Pure and Applied Algebra vol:214 issue:6 pages:841-849
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 oil X (the real gonality of X). Then we can find oil 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 g = 2(n) - 2 (n >= 1) we prove that k(R) <= 8/2 + 1 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. (C) 2009 Elsevier B.V. All rights reserved.