I'm studying Miranda's book [Algebraic curves and Riemann surfaces]. I have a question about the base-point free linear systems. In chapter 5, proposition 4.15 of the text, the author explained the following fact:
Let $Q\subset |D|$ be a base-point free linear system of dimension $n$ on a compact Riemann surface (i.e., $\forall p\in X$, $\exists E\in |D|,~ E(p)\neq 0$. ). Then there exists a holomorphic map $\phi: X\rightarrow \mathbb{P}^n$ such that $|\phi|=Q$.
The proof is as follows: Since $|D|$ has a natural bijective correspondence with $\mathbb{P}(L(D))$, we suppose that $Q$ corresponds to the pullback subspace $V\subset L(D)$ of dimension $n+1$. Now it suffices to choose a basis of $V$, say $\{f_0,...,f_n\}$; the map given by $\phi:=[f_0:...:f_n]$ is what we want here.
The author gives a rather concise sketch here and says "we've been running around these ideas enough now that the proof of the proposition is almost easier than the statement".
But I'm puzzled here: this concrete construction doesn't use the condition "$Q$ is base-point free". That is, for an arbitrary linear system you can still repeat this process to obtain a certain $\phi$. (Am I right? I'm not very sure!) The conclusion $|\phi|=Q$ is impossible now, but I can't see why it doesn't hold. Can anybody point out what I have missed? (By the way, I am also curious about the geometric picture/intuition of these abstract notions: linear systems, base points, etc.) Thanks a lot in advance!