We shall work over the complex number field C. Let X be a non-singular projective curve of genus g. We always assume that it is non-hyperelliptic and sometimes identify it with its canonical image in PH(X,KX). By Max Noether’s theorem, the canonical ring of X is generated in degree one and the canonical map is a projectively normal embedding. Furthermore, a well-known theorem of Enriques-Petri states that X is cut out by hyperquadrics if Cliff(X), the Clifford index of X, is bigger than one. Then Mark Green [4] conjectured that the non-vanishing of a certain higher syzygy can be characterized by the Clifford index, which has been verified in many cases. From these, we learn that Cliff(X) reflects the algebraic structure on X better than the gonality gon(X), while two invariants are almost equivalent [2]. For a non-negative integer k, we denote by X the k-th symmetric product of X whose points are considered as effective divisors of degree k on X. When k = 0, we understand that X is one point corresponding to the zero divisor. Let Dk be the open subset of X consisting of effective divisors D which spans scheme theoretically a (k − 1)-plane 〈D〉 in PH(X,KX). For D ∈ Dk, we put KX,D = KX − [D], where [D] denotes the line bundle associated to D. We let ΦKX,D : X → PH(X,KX,D) be the rational map associated to the complete linear system |KX,D|. In this article, we consider two kinds of uniformity questions with respect to Dk