Let X a smooth projective curve over $\mathbb{C}$.
We fix $d$ distinct closed points $x\_{1},\dots,x\_{d}$.


Can we find a finite surjective morphism $\pi:X\rightarrow\mathbb{P}^{1}$
and local uniformizers on $\mathbb{P}^{1}$, $t\_{1},\dots,t\_{d}$


such that $\forall i, k[t\_{i}^{-1}]\subset k[X-x\_{i}]$




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finite surjective morphism to the projective line
No. This condition implies that $t\_i$, pulled back to $X$, vanishes only at $x\_i$. Thus, $x\_i$ is the entire fiber of the point where $t\_i$ vanishes, so if $n$ is the degree of the map, then $\mathcal O(1) = \mathcal O(n x\_i)$.


Thus $\mathcal O(nx\_i)=\mathcal O(nx\_j)$, so $x\_i-x\_j$ is an $n$-torsion divisor class, for all $i$, $j$.


To show this is impossible, it suffices to find two points $x\_1$ and $x\_2$ on a curve, such that $x\_1-x\_2$ is not a torsion divisor. To do this, let $E$ be an elliptic curve, let $x\_1$ be the identity, and let $x\_2$ be a non-torsion point. Then $x\_1-x\_2$ is not a torsion divisor class.


