# Linear systems separating points

Is it easy to find an example of a complete linear system on a smooth projective curve (say over $$\mathbb C$$) which separates points but which is not an embedding?

(for just a linear system, one can take the linear system induced by a linear projection (in an embedding) from a point which is not on any secant line of the curve but lies on a tangent line of the curve).

Take for $$X$$ a trigonal curve of genus $$\geq 5$$; that is, $$X$$ carries a unique degree 3 pencil $$P$$. Suppose some divisor of $$P$$ is of the form $$p+2q$$. Consider the linear system $$|K-p|$$. If $$r,s$$ are two distinct points of $$X$$, we have $$h^0(p+r+s)=1$$ by unicity of the $$g^1_3$$, hence $$h^0(K-p-r-s)=h^0(K-p)-2$$ by Riemann-Roch; thus $$|K-p|$$ separates $$r$$ and $$s$$. But $$h^0(K-p-2q)=h^0(K-p)-1$$, which means that the associated map is not an embedding at $$q$$.
I believe that on any non-rational and non-hyperelliptic curve , the complete linear system $$L = K_X(2p)$$ will work. In such a case $$2p - p_1 -p_2$$ will always be a non-trivial divisor of degree zero unless $$p = p_1 = p_2$$ and hence $$h^0(L) = g+1$$ and $$h^0(L(-p_1 -p_2) ) = g-1$$ unless as stated $$p = p_1 = p_2$$.