Let $X$ be a smooth projective surface $S$ in $\mathbb P^3$. Let $C$ be a smooth hyperplane section of $S$ and $D$ be a non-zero divisor on $S$. Consider the short exact sequence : $0 \to \mathcal O_S(D-(m+1)C) \to \mathcal O_S(D-mC) \to \mathcal O_C(D-mC) \to 0$. In this context my question is the following :
If we know that $h^0(\mathcal O_C(D)) <h^0(\mathcal O_C(mC))$, then can we say $h^0(O_C(D-mC))=0$?
Any remark or insight from anyone is appreciated.
If $D'$ is an effective divisor, the morphism $$ H^0(\mathcal{O}(D'')) \to H^0(\mathcal{O}(D' + D'')) $$ given by $D'$ is injective for any divisor class $D''$; in paricular $$ \dim H^0(\mathcal{O}(D'')) \le \dim H^0(\mathcal{O}(D' + D'')). $$ Now apply this observation for $D'' = mC$ and $D'$ any divisor in $|D - mC|$.