Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 25 / 6 / a / Solution

The adjunction formula on the degree-$d$ projective hypersurface gives $K_X=(d-4)H$. Since $C$ is a smooth rational curve, adjunction on $C$ gives $(K_X+C)\cdot C=2g(C)-2=-2$. Its degree is $H\cdot C=a$, so the smooth rational-curve self-intersection on a projective hypersurface formula is