# Fulton's Algebraic Curves Exercise 8.13

This exercise is from Fulton's Algebraic Curves, exercise 8.13, page 101. The exercise reads as follows

Suppose $$l(D)>0$$ and let $$f \neq 0, f \in L(D)$$. Show that $$f \notin L(D-P)$$ for all but a finite number of $$P$$. So $$l(D-P)=l(D)-1$$ for all but a finite number of $$P$$.

Here $$L(D)$$ represents the linear system (or series if you prefer) of some divisor $$D$$, while $$l(D)$$ denotes the dimension of this vector space.

My Attempt

As $$l(D) > 0$$, without loss of generality we may assume that $$D$$ is effective, so we can write $$D = \sum_{Q \in I} n_QQ$$ where $$I$$ is some finite set of points and all of the $$n_Q > 0$$. My guess is that we require $$P \in I$$ for the proposition to hold. Suppose instead that $$P \notin I$$. Now for $$f \in L(D)$$ we see that $$val_P(f) \geq 0$$, while for $$g \in L(D-P)$$, we need $$val_P(g) \geq 1$$. Additionally, if $$P \in I$$, we have $$val_P(f) \geq -n_P$$ and for any $$g \in L(D-P)$$ we have $$val_P(g) \geq 1-n_P$$.

In both of these cases it seems to me that there could be some $$f$$ which could be an element of $$L(D-P)$$ for any chosen $$P$$. Could someone show me what I'm missing with this problem, I feel like it should be straightforward but I can't figure out what to try next.

Write $$D = \sum_{Q \in I} n_Q Q$$ for some finite set of points $$I$$. Let $$f \in L(D)$$.
Assume that there is some infinite set $$J$$ such that for all $$P$$ in $$J$$, $$f \in L(D - P)$$. For each point $$P \in J - I$$, we have that $$n_P = 0$$. Since $$f \in L(D - P)$$, we have $$v_p(f) + n_P - 1 \geq 0$$, i.e; $$v_P(f) \geq 1$$ for all $$P \in J - I$$. This is impossible as $$v_P(f)$$ is nonzero only for finitely many points.
Thus $$f \notin L(D - P)$$ for all but a finite amount of $$P$$.