Journal of the
Korean Mathematical Society JKMS

For a smooth projective irreducible algebraic curve $C$ of odd gonality, the maximal possible dimension of the variety of special linear systems $W^r_d(C)$ is $d-3r$ by a result of M. Coppens et al. [4]. This bound also holds if $C$ does not admit an involution. Furthermore it is known that if $\dim W^r_d(C)\ge d-3r-1$ for a curve $C$ of odd gonality, then $C$ is of very special type of curves by a recent progress made by G. Martens [11] and Kato-Keem [9]. The purpose of this paper is to pursue similar results for curves of even gonality which does not admit an involution.

Keywords: algebraic curves, linear series, gonality, Brill-Noether theory

MSC numbers: 14H51, 14C20