Let $X$ be a nonsingular curve over an algebraically closed field $k$ (by curve over $k$, I mean an integral separated scheme of finite type over $k$ which has dimension 1). Let $P$ and $Q$ be distinct closed points of $X$.
Question: Does there exist a rational function $f$ on $X$ satisfying both of the following two properties:
- $f$ has a simple zero at $P$;
- $f_Q\in \mathcal{O}_Q^*$, i.e. $f$ does not have a pole or a zero at $Q$.
Equivalently, does there exist a rational function $f$ on $X$ such that the principal (Weil) divisor $\operatorname{div}(f)$ given by $f$ is of the form $$ \operatorname{div}(f) = P + D, $$ where $D\in \operatorname{Div}(X)$, and $P,Q\notin \operatorname{Supp}(D)$.
Equivalently-equivalently, is $P$ linearly equivalent to a divisor $D$ with $P,Q\notin\operatorname{Supp}(D)$?
Motivation: I'd like to be able to say that, given closed points $P_1,\ldots,P_n\in X$, any divisor $D$ on $X$ is linearly equivalent to one without $P_1,\ldots,P_n$ in its support.
EDIT: My background is Chapters I and II in Hartshorne, so I'd like to be able to prove this without, e.g., Riemann-Roch.
Embed $C$ in a projective space. Take a generic linear section through $P$ that doesn't contain $Q$. Divide by a generic linear section that contains neither $P$ nor $Q$. This is down to earth and needs only Bertini.
UPD: In response to a completely reasonable comment, here is how to choose an embedding. We may assume that $C$ is irreducible. Cover the curve by finitely many affine opens $C=\cup U_i$. Think of each $U_i$ as of a smooth curve in $\mathbb{A}^{n_i}\subset\mathbb{P}^{n_i}$ and take the closure $C_i$. Now, we have a natural morphism $U_i\to C_i$ that extends to $f_i:C\to Y_i$ (valuative criterion of properness [H, Chapter II]). Consider the diagonal map sending $C\to \Pi_i C_i$ this would give the desired embedding into a projective curve (take the closure of the image).