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 $\phi:X \rightarrow \Bbb P^n$, $x \mapsto [f_0(x): \dots : f_n(x)]$. 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 $\text{div}(g)+D$, where $D$ is defined as $-\min_i\{\text{div}(f_i)\}$ 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.
Thank you in advance.
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.