# intersection on geometrically ruled surface

$S$ : a geometrically ruled surface (i.e., $S$ is a $\mathbb{P}^1-$bundle over a smooth curve)

$F$ : a fibre.

$A$ : an ample divisor on $S$

Then $((K_X+2F).A)(A.F)-(K_X.F-F^2)A^2\ge 0$.

I tried to calculate this.

Since $F$ is a fibre, $(K_X.F-F^2)=-2$ by adjunction formula. So $((K_X+2F).A)(A.F)-(K_X.F-F^2)A^2=((K_X+2F).A)(A.F)+2A^2$. Since $A$ is ample, $A.F,A^2\ge0$. But I don't know anything about $(K_X+2F).A$.