Let X be an integral Gorenstein projective curve with g:=pa(X) $geq$ 3. Call $G^r_d$ (X,**) the set of all pairs (L,V) with L$epsilon$Pic(X), deg(L) = d, V $subseteq$ H^0$(X,L), dim(V) =r+1 and V spanning L. Assume the existence of integers d, r with 1 $leq$ r$leq$ d $leq$ g-1 such that there exists an irreducible component, , of $G^r_d$(X,**) with dim($Gamma$) $geq$ d - 2r and such that the general L$geq$$Gamma$ is spanned at every point of Sing(X). Here we prove that dim( ) = d-2r and X is hyperelliptic.