Let $X$ be a smooth projective complex surface and $C$ a smooth curve on it. How can one conclude that $(C,C)=-1$ provided that $(C,C)+(K_X,C)<0$?
By adjunction formula, $(C,C)+(K_X,C)=2g(C)-2$, we have $C\cong \mathbb{P}^1$, but I don't see why its self-intersection number is $-1$.