Let $C$ be smooth algebraic curve of degree $7$, embedded in $\mathbb{CP}^2$ which is a curve defined by one polynomial in two variables. Let $D = p_1+p_2 + \cdots +p_6$ and let all $6$ points lie on the same complex line.
What is $l(D)$, the dimension of vector space of meromorphic functions on $C$, which have no simple poles outside D. Let $K$ stand for canonical divisor.
$l(K)=g$, $g = \frac{5 \cdot 6}{2} =15$, $deg D = 6$, $l(0)=1$,
By Riemann-Roch theorem we get that $l(D) = l(K-D) - 8 \leq 7$.
The situation is that we have $L(0)$ which is one-dimensional, and as we add points to the divisor we get more and more functions, so the dimension rises by no more than 1 with each point. But since all the points are on the same complex line, we can't be sure that every time the dimension rises. How should I use the fact that all the points are on the same plane?
I have got this task on my Riemann surfaces exam last spring. I am allowed to use any theorem or fact.