@DonuArapura As far as I know $C \cdot D \geq \#(C \cap D)$ (since it is the sume of the intersection multiplicities at points of $C \cap D$), so I do not understand how $ \# (C \cap D)$ can be inifnite but not $C\cdot D$. On the other hand, when you say that "$C \cap D$ might be infinite, even in the plane", I guess you make reference to the projective plane, but here we have $\# (C \cap D) \leq nm$. You're answer doesn't convince me, but thank you anyway.