= Rational functions with prescribed poles
For finitely many distinct points $P_i$ on a smooth projective genus-$g$ curve, choose $N\ge2g+1$. The <Riemann-Roch theorem> gives $\ell(NP_i)-\ell((N-1)P_i)=1$, so choose $f_i$ with a sole pole of exact order $N$ at $P_i$. Then $\sum_i f_i$ has exactly those poles with that order: all other summands are regular at each $P_i$, so no principal part can cancel. This is an explicit construction rather than an unspecified generic choice of section.
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