Self-intersection after blowing up points on a smooth curve
= Self-intersection after blowing up points on a smooth curve
Blowing up $r$ distinct smooth points of a smooth curve $C$ gives a strict transform $C'$ isomorphic to $C$ with
$$
(C')^2=C^2-r.
$$
Choosing $r>C^2$ turns a curve of positive self-intersection into one of negative self-intersection without changing its abstract isomorphism type.