Linear pencils on real algebraic curves

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 XX denote a real curve of genus gg with real points and let k(R)k(R) be the smallest degree of a pencil on XX (the real gonality of XX). Then we can find on XX a base point free pencil of degree g+1g+1 (resp. gg if XX is not hyperelliptic, i.e. if k(R)>2k(R)>2) with an assigned geometric behaviour w.r.t. the real components of XX, and if g=2n−2(n≥1) we prove that k(R)≤g2+1 which is the same bound as for the gonality of a complex curve of even genus gg. Furthermore, if the complexification of XX is a kk-gonal curve (k≥2)(k≥2) one knows that k≤k(R)≤2k−2k≤k(R)≤2k−2, and we show that for any two integers k≥2k≥2 and 0≤n≤k−20≤n≤k−2 there is a real curve with real points and kk-gonal complexification such that its real gonality is k+nk+n.