Base-point-free linear systems (elementary?) property

I'm having troubles solving exercise K on page 167 of the book "Algebraic curves and Riemann surfaces" of Miranda.

The question is the following one : Let Q be a base-point-free linear system, let p_1, ...,p_m be some points on the Riemann Surface X. Show that there is a divisor D in Q without any p_i in its support.

I tried the following :

Since Q is a base-point-free linear system, it corresponds to some linear system |phi| associated to some holomorphic map phi:X \rightarrow P^n: x \mapsto [f_0(x): \dots : f_n(x)]. Here the f_i's denote some meromorphic functions. We have a very clear discription of such a system, namely any divisor in it can be written as

div(g)+D,

where D is defined as -min_i{div(f_i)} and g is a linear combination of the f_i's.

Since we're dealing with a base-point-free system, for any i, there is a divisor E_i in Q such that E_i(p_i)=0.

But now I'm stuck, I tried some combinations of the E_i's to get the desired divisor, but I don't find it.

Remarks : I don't know much about sheaves and schemes, so I can't use it. And I'm very sorry for the notation above, I'm new to this site and I haven't figured out yet how to get nice notations.

This may be cheating but here goes, since $D$ is base point free $\dim L(D) < \dim L(D+p)$ for all $p$. You want to show that $L(D) \neq \cup_{i} L(D+p_i)$ but this should be obvious since a vector space cannot be the union of a finite number of spaces of smaller dimension.
• Wnat I mean to say, I am not sure I am using the right notation, but if you consider all divisors which contain $p$ (that is you add $p$ as a base point) then you get a system of lower dimension. The question is for a divisor which does not lie in a finite number of such subsystems. Whats the notation for adding $p$ as base point ? $D+p$ ? – Rene Schipperus Jun 8 '14 at 13:58