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.