Special Linear Systems on Algebraic Curves

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Let C be a smooth algebraic curve. Assume that C does not admit a double covering to another curve when C is an eve-gonal curve. We proved that if the dimension of a family of line bundles with degree d in which d is less than g-5, is more than d-3r-2, then C is a d-gonal curve with d【less than or equal】6, a 4-sheeted covering of an elliptic curve or a plane curve of degree 8. This is our first result. Next, in case 6-gonal we can prove that except a triple covering of an elliptic curve, the dimension of a family of line bundles with degree d in which d is less than g-5, should be less than d-3r-3, and in case 4-sheeted covering of an elliptic curve, the same conclusion holds.