In Section 1, we investigate the problem of the existence of base point free pencils of relatively low degree on a double covering X of genus g of a general curve C of genus q > 0. Such problem is classical and the picture is rather well known for the degree range close to the genus g. On the other hand, by a simple application of the Castelnuovo-Severi inequality one can easily see that there does not exist a base point free pencil of degree less than or equal to g-2 q other than pull-backs from the base curve C; while for the degree beyond this range not many things have been known about the existence of such a pencil which is not composed with the given involution. Here we prove the following result.
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