Both $B$ and $C$ are curves contained in the intersection of $S_1$ and $S_2$, but $B$ is contained in $V$ and $C$ is not. Moreover, if $T$ is a (not nec. reduced) surface contained in the intersection of $S_1$ and $S_2$ and such that $T_{|V}=B$ then I believe that the inequality I wrote is correct. In general, $T$ might not exist (as $S_1$ and $S_2$ might be irreducible). I was wondering if the inequality still holds. I am not sure if this answer your questions.