Fix a generic plane curve $D$ in ${\bf P}^2$, what is the minimal number of intersections one can obtain by intersecting $D$ with another curve $C$? Here we count the number of intersections set-theoretically; otherwise, the number is constant and given by Bezout. If $C$ is a line, it is not hard to figure out that the minimum is achieved by a bitangent or flex line. Surprisingly or not, this is also the minimum regardless of the degree of $C$.
I will discuss the connection of this problem to the Kobayashi conjecture in complex geometry and also the generalization of the above result to rational surfaces other than ${\bf P}^2$.