Let \(\pi \colon \tilde{X} = \Bl_p X \to X\) be the blowup of a smooth surface at a point, with exceptional curve \(E \cong \PP^1\). Then \[ \Pic \tilde{X} = \pi^* \Pic X \oplus \ZZ E , \] and the intersection numbers are \[ \pi^*C \cdot \pi^*D = C \cdot D , \qquad \pi^*C \cdot E = 0 , \qquad E^2 = -1 . \] For the canonical divisor, \[ K_{\tilde{X}} = \pi^* K_X + E , \qquad K_{\tilde{X}}^2 = K_X^2 - 1 , \] and \(\chi(\OO)\), \(p_g\) and \(q\) are unchanged.
(Hironaka resolution (characteristic zero))
Let \(k = \CC\).
For every projective variety \(Y\) there exist a smooth projective variety \(Y'\) and a birational morphism \(\varphi \colon Y' \to Y\) such that \(\varphi\) is an isomorphism over \(\reg Y\).
For every smooth projective variety \(X\) and projective subvariety \(Y \subset X\) there exist a smooth projective variety \(W\) and a birational morphism \(\varphi \colon W \to X\) such that \(\varphi^{-1}(Y)\) is a divisor with simple normal crossings in \(W\).
If \(Y \subset X\) as in (b) is a hypersurface, there is a unique irreducible component \(Y'\) of \(\varphi^{-1}(Y)\) dominating \(Y\), and \(\varphi\vert_{Y'} \colon Y' \to Y\) is a desingularization as in (a); it is called an embedded desingularization of \(Y\).
In (a) and (b) the morphism \(\varphi\) can be chosen as a composition of blowups with smooth centres.
In particular every projective variety admits a desingularization by a sequence of blowups with smooth centres, and every subvariety can be made SNC after such a sequence.
(Strict transform)
For \(C \subseteq X\) passing through \(p\) with multiplicity \(m\), the strict transform \(\tilde{C}\) is the closure of \(\pi^{-1}(C \setminus p)\), and \[ \pi^* C = \tilde{C} + mE , \qquad \tilde{C}^2 = C^2 - m^2 , \qquad p_a(\tilde{C}) = p_a(C) - \binom{m}{2} . \]
\(E^2 = -1\) is the fact the rest hangs on, and it follows from \(\pi^*C \cdot E = 0\) applied to a curve through \(p\): writing \(\pi^*C = \tilde{C} + E\) for a smooth \(C\) gives \(0 = \tilde{C}\cdot E + E^2 = 1 + E^2\).
The genus drop is the tool for resolving plane curve singularities: blowing up an ordinary \(m\)-fold point removes \(\binom{m}{2}\) from the arithmetic genus, which matches the delta invariant, so repeated blowups compute the geometric genus. A node has \(m=2\) and drops the genus by one.
\(K^2\) decreasing by one while \(\chi(\OO)\) stays fixed is the numerical fingerprint of a blowup, and it is what Noether’s formula balances: \(c_2\) goes up by one.
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