Suppose $C_1,C_2$ are embedded complex curves in a complex surface $S$, and $C_1,C_2$ have no common component. Assuming $C_1$ and $C_2$ intersect transversally, the intersection number $C_1\cdot C_2$ is just the cardinality of $C_1\cap C_2$, because they should intersect positively. Now suppose $C_1$ and $C_2$ intersect (not necessarily transversally) at finitely many points , say $n$. Then is it true that $C_1\cdot C_2\geq n$, with equality iff $C_1$ intersects $C_2$ transversally?
(Actually I am in a situation that $C_1\cap C_2$ is $n$ points and want to show that $C_1,C_2$ intersect transversally. If this result is true then I may conclude that $C_1,C_2$ intersect transversally, but I'm not sure about this question.)