  We study the following question: fix a sufficient general curve D of degree d
in P^2, what is the least number of intersections between D and an irreducible
curve of degree m? G. Xu proved this number i(d, m) is at least d - 2 for all
m. This problem can be regarded as the algebraic part of Kobayashi conjecture
on the hyperbolicity of P^2 D. We first improved Xu's bound with m fixed and
then generalized his result to rational ruled surfaces.
