Abstract
For a smooth projective irreducible algebraic curve C of odd gonality, the maximal possible dimension of the variety of special linear systems Wdr(C) is d-3r by a result of M. Coppens et al. Furthermore it is known that if the maximum dimension of W(C) for a curve C of odd gonality is attained then C is of very special type of curves by the recent progress made by G. Martens. The purpose of this paper is to chase an extension of the result of G. Martens; if dim W(C)=d-3r-1 for a curve C of odd gonality for some d≤g-4 and r≥1, then C must be either a smooth plane sextic, a pentagonal curve of bounded genus or a smooth plane octic.
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