Let $X$ be a compact Riemann surface, let $D$ be a divisor on $X$.
Consider the Riemann-Roch space $L(D)$.
Take some point $p \in X$ such that $p$ is not in $D$ (that is, the support of $p$ at $D$ is $0$).
Could it happen that $f(p) = 0$ for all $p \in L(D)$? I guess this means that $L(D) = L(D-p)$. Could this happen? Is there a name for such points $p$ and a characterization of them?