(Log resolution)
Let \(X\) be a variety and \(D \subseteq X\) a closed subset of codimension at least \(1\), possibly empty. A log resolution of the pair \((X, D)\) is a proper birational morphism \(f \colon Y \to X\) such that \(Y\) is smooth, the exceptional locus \(\operatorname{Exc}(f)\) is a divisor, and \(\operatorname{Exc}(f) \cup f^{-1}(D)\) is a simple normal crossing divisor.
(Hironaka)
Over a field of characteristic \(0\), every pair \((X, D)\) has a log resolution, which can be taken to be a composite of blowups along smooth centres and an isomorphism over the open set where \(X\) is smooth and \(D\) is a simple normal crossing divisor.
For the cuspidal curve \(D = V(y^2 - x^3) \subseteq \AA^2 = X\), one blowup at the origin makes the strict transform smooth but tangent to the exceptional curve \(E_1\). A second blowup leaves the strict transform and the two exceptional curves passing through one point, which is not a normal crossing. A third blowup at that point gives a log resolution: the total transform is a simple normal crossing divisor.
Dependencies
D-VARNCROSSNormal crossing singularitiesusesD-VARBLOWThe blowup of a variety at a pointuses