Rational functions over variety X

Question

I 'm trying to solve this exercise of Fulton Algebraic Curves:

Let $D=\sum n_P P$ be an effective divisor, $S=\{P \in X: n_P>0\}$, $U=X\setminus S$. Show that $L(rD)\subset \Gamma(U,\mathcal{O}_X)$. Where $L(E)=\{f \in K : \text{ord}_P(f) >-n_P ,\forall P \in X\}$ (The linear vectorial space of the rational functions with pole multiplicity worst at $n_P$).

But I have no idea ... any suggestions to get started? Thanks.

Answer 1

Here's a hint:

By definition, the $\Gamma(U,\mathcal O_X)$ are the regular functions on $U$ - in other words, the functions on $X$ that have no poles on $U$.

And here's a stronger hint, hidden for you (mouse-over if you really want to see):

Also, $L(rD)$ consist of the functions $f$ such that $div(f)+rD \geq 0$. This happens only if $f$ has at most poles of order $rn_P$ at $P$ and nowhere else. In particular, the elements of $L(rD)$ are are regular functions on $U$.