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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 $C$?

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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 $\frac{\beta(\phi(p_i))}{\lambda(\phi(p_i))} = a_i$, $i=1,2$.

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  • $\begingroup$ Sorry if this is a standard result about embeddings associated to very ample divisors, but why does the existence of the hyperplane with those properties follow? $\endgroup$ – Peter Xu Jun 2 '15 at 4:23
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    $\begingroup$ Have a look to the following (after doing that let me know if it is now clear to you): math.stackexchange.com/questions/1197197/… $\endgroup$ – Holonomia Jun 2 '15 at 9:46
  • $\begingroup$ Oh, I see. Thank you! $\endgroup$ – Peter Xu Jun 2 '15 at 14:24

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