Smooth rational-curve self-intersection on a projective hypersurface
= Smooth rational-curve self-intersection on a projective hypersurface
{title2=$C^2=-2-(d-4)a$}
For a degree-$d$ smooth surface $X\subset\mathbb P^3$ and a smooth rational curve $C$ of degree $a$, the <canonical bundle> is $K_X=\mathcal O_X(d-4)$. The <adjunction formula> gives $(K_X+C)\cdot C=-2$, hence $C^2=-2-(d-4)a$. In particular a line has self-intersection $2-d$.