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.

What does the self-intersection number calculation in homology say?

Update, corrected: 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(C1)(C2)

which is just 2xy. Perhaps this will shed some light on the subject.

What does the self-intersection number calculation in homology say?

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.