I have been reading Rick's Miranda book on Riemann surfaces and now he states
Let $Q \subset |D|$ be a base point free linear system of (projective ) dimension $n$ on a compact Riemann surface $X$. Then there is a holomorphic map $\phi: X \rightarrow \mathbb{P}^n$ such that $Q=|\phi|$.
Now to prove this we take the associated vector space $V \in L(D)$ that is associated with $Q$ and choose a basis $f_0,…,f_n$ for $V$ and make $\phi=[f_0,…,f_n]$, and he claims $|\phi|=Q$. My question is doenst he need to see that if we define $D'=-min\{div(f_i)\}$ ,where this $f_i$ run trought the basis, that $D' =D$ ? Because when he is defining a linear system from a holomorphic map this is the construction that he makes for the divisor, and it will have an impact on the way the linear system is defined trough this map. Thanks in advance.
Best Answer
Ok I belive i got it we do need to verify that but it is going to be true because $Q$ is base-point free linear system so $\forall p \in X$ we have that $V$ is not contained in $L(D-p)$.