Could you please give me a nice example of the following fact? Let $X$ be an algebraic surface, let $D$ be a divisor on it and let $C$ be a curve not included in the support of $D$. Then $$ \dim \frac{ \mathcal{L}(D)}{\mathcal{L}(D-C)} \leq D \cdot C +1 $$ where on the right hand side we have the intersection multiplicity.
I have tried this cases but I was not able to figure them out: a line (as $D$) and a smooth conic/cubic (as $C$) on a plane.
Is it also possible to give some examples of this on quadric surfaces?
Thanks in advance!
Say $C$ is a complete irreducible curve, let $k(C)$ its function field,
Think to $C\cap D =\sum_{j=1}^J n_j P_j$ as a divisor on $C$.
If $f\in \mathcal{L}(D)$ then its image in $k(C)$ is in $\mathcal{L}_C(\sum_{j=1}^J n_j P_j)$, that space is at most $\sum_{j=1}^J n_j \deg(P_j)+1=C.D+1$ dimensional.
The kernel of the map $\mathcal{L}(D)\to k(C)$ is $\mathcal{L}(D-C)$.