What does the self-intersection number calculation in homology say?
Update: Here's what I mean. Consider a curve that has class xC1 + yC2, then its square is
x^2 (C1)^2 + y^2 (C2)^2 + 2xy
which has the asymptotics of negative infinity when x or y are large. So you just have to show that you have many curves with x and y large, which I think somehow follows from standard statements.