On the variety Wdr(C) whose dimension is at least d−3r−2

Let C be a smooth projective algebraic curve which is not a curve of even gonality admitting an automorphism of order 2. Under this rather mild restriction on C, we establish a stronger dimension theoretic statement concerning the variety of special linear systems; we prove that if dim Wdr(C)greater-or-equal, slantedd−3r−2greater-or-equal, slanted0 for some d,r>0 with dless-than-or-equals, slantg−5 and ggreater-or-equal, slanted21, then C is a k-gonal curve with kless-than-or-equals, slant6, a 4-sheeted covering of an elliptic curve or a plane curve of degree 8.