Exists rational function on curve in $\mathbb{CP}^2$ such that pole of order $2g + 2$?

Question

Let $C \subset \mathbb{CP}^2$ be a nonsingular curve of degree $d$, and $p_1$, $p_2$, $q$ distinct points in $C$. For any $a_1$, $a_2 \in \mathbb{C}$, does there necessarily exist a rational function $f$ on $C$ with $f(p_i) = a_i$ and a pole of order $2g + 2$ at $q$, where $g$ is the genus of $C$?

Answer 1

The answer is yes. You have to consider the divisor $D = (2g + 2)q$. Since $deg(D) = 2g + 2 > 2g$ it follows that $D$ is very ample. This means that there is an embedding $\phi: C \to \mathbb{P}^r $ in a projective space $\mathbb{P}^r$ with $r \geq 2$ and a hyperplane $H \subset \mathbb{P}^{r}$ passing exactly through the image $\phi(q)$ which has contact of order $2g + 2$ with the image $\phi(C)$. The hyperplane $H$ is the zero set of a linear form $\lambda$. Consider now the rational functions on $\phi(C)$ obtained as quotients $\frac{\beta}{\lambda}$ where $\beta$ is an arbitrary linear form. All such rational function have a pole of order $2g + 2$ at $\phi(q)$. Now by using that $r \geq 2$ you can find $\beta$ such that $\frac{\beta(\phi(p_i))}{\lambda(\phi(p_i))} = a_i$, $i=1,2$.