Let $X$ be a compact surface of genus $g\ge 2$. I want to show that there exists $P_1,\dots,P_g$ distinct points on $X$, such that a holomorphic differential $\omega\in H^0(X,\Omega_X^1)$ must be zero if $\omega(P_j)=0,j=1,2,\dots,g$.
I know a fact: Let $X$ be a compact surface of genus $g\ge 1$, $g=1$ if and only if there exists $\omega\in H^0(X,\Omega_X^1)$ such that $\omega(P)\ne 0 $ for all $P\in X$.
Let $K$ be the canonical class of $X$. By Riemann-Roch theorem I know $d(K)=2g-2$. So when $g\ge 2$, $div(\omega)=d(K)=2g-2> 0$ for all $\omega\in H^0(X,\Omega_X^1)$. Let $(w)=\sum_{P} v_P(\omega)P$, then $\sum_{P} v_P(\omega)=2g-2$. But I don't know whether these facts can help me solve the problem.
I'm confused on how to prove it. Any ideas?