Let \(X\) be a smooth projective surface, \(\pi: \Bl_p X \to X\) the blowup at a point, \(E\) the exceptional curve. Then \[ E \cong \PP^1, \qquad E^2 = -1, \qquad K_{\Bl_p X} = \pi^* K_X + E . \] In dimension \(n\) the last formula reads \(K_{\Bl_p X} = \pi^* K_X + (n-1)E\).
(of the self-intersection)
Pullback preserves intersection numbers, and \(\pi^* A . E = 0\) for every divisor \(A\) on \(X\), since \(\pi_* E = 0\). For \(C \subseteq X\) a curve smooth at \(p\), the proper transform satisfies \(\pi^* C = \tilde{C} + E\) and \(\tilde{C}\) meets \(E\) transversally in the one point recording the tangent direction of \(C\). Intersecting with \(E\): \[ 0 = \pi^* C . E = (\tilde{C} + E).E = \tilde{C}.E + E^2 = 1 + E^2 . \]
An alternative geometric computation uses two lines in \(\PP^2\): when they meet transversally away from \(p\), \(\pi^* L_1 . \pi^* L_2 = L_1 . L_2 = 1\); moving them to meet at \(p\) gives \(\pi^* L_i' = \tilde{L}_i' + E\), while the proper transforms miss each other, so expanding gives \(1 = 0 + 1 + 1 + E^2\). Both arguments run on the same two facts, \(\pi^*A.E = 0\) and \(\pi^*C = \tilde{C} + E\).
The consequence is Castelnuovo’s criterion — a smooth rational curve with \(E^2 = -1\) on a surface is the exceptional curve of a blowup and can be contracted — which is what makes minimal models of surfaces possible. The canonical formula is the reason blowing up strictly increases \(K^2\)-deficiency: \(K^2\) drops by one each time, and \(p_a\) and \(p_g\) do not change.