On the complement of a generic curve in the projective plane

Let C be a very generic smooth curve of degree d in the complex projective plane. In this paper, we show that for any curve D in the projective plane that does not contain C, the set C ∩ D contains at least d - 2 distinct points. Moreover, this bound is sharp when d ≥ 3. Our motivation for the study of this problem comes from the problem of hyperbolicity of the complement of a plane curve. As an application of our result, we give another proof of the following fact: if C is a very generic smooth curve of degree d ≥ 5 in the complex projective plane, then the intersection of the complement of C in the projective plane with any plane curve D which does not contain C is hyperbolic.