Linear series on k-gonal curves

Here we prove the following result. Theorem 1.1. Let X be an integral projective curve of arithmetic genus g and k≧ ≧4 an integer. Assume the existence of L ∈ Pick(X) with h0(X, L)=2 and L spanned. Fix a rank 1 torsion free sheaf M on X with h0(X,M)=r+1≧2, h1(X, M)≧2 and M spanned by its global sections. Set d:=deg(M) and s:=max {n≧0:h0 (X, M ⊗(L*)⊗n)>0}. Then one of the following cases occur: (a) M ≅ L⊗n;(b)M is the subsheaf of ωX⊗(L*)⊗t, t:=g-d+r-1, spanned by H0(X, ωX⊗(L*)⊗t); (c)there is a rank 1 torsion free sheaf F on X with 1≤h0(X, F)≤k-2 such that M≅L⊗s⊗F. Moreover, if we fix an integer m with 2≤m≤k-2 and assume r≠(s+1)k-(ns+n+1) per every 2≤n≤m, we have h0(X, F)≤k-m-1. We find also other upper bounds on h0 (X, F). © 2001 Università degli Studi di Ferrara.