Intersection formula for blowing up a surface

If $\pi :\tilde{X}\to X$ blows up a smooth point with exceptional curve $E$, thenIf a curve $C$ has multiplicity $m$ at the center, its strict transform is $\tilde{C}={\pi }^{\ast }C-mE$ and ${\tilde{C}}^2=C^2-m^2$.

Every nonisomorphic birational morphism from a smooth projective surface to ${\mathbb{P}}^2$ factors into point blowups. If $H$ is the pullback of a line and $E$ is the total transform of an exceptional divisor from one factor, then $H^2=1$, $E^2=-1$ and $H\cdot E=0$, so $(H+E{)}^2=0$.

Blowing up $r$ distinct smooth points of a smooth curve $C$ gives a strict transform $C^{\prime }$ isomorphic to $C$ withChoosing $r>C^2$ turns a curve of positive self-intersection into one of negative self-intersection without changing its abstract isomorphism type.