Let $X$ be a non compact complex surface non projective and non algebraic, and let $S$ be compact Riemann surface embedded in $X$ ( i mean that $S$ is a compact complex sub variety of $X$ of dimension 1 ).
Let $f$ be a meromorphic function on $X$ ( i mean $f: X \longrightarrow \mathbb{P}^1=\mathbb{C} \cup \infty $ is holomorphic )
if $(f)$ is the principal divisor associated to $f$, is it true that $(f).S = 0$ ? if not could you give a counter example ? ( i mean by $(f).S$ the intersection between two divisor since $S$ is compact this intersection does make sense i think )
thanks in advance for your help.
