Resolution of singularities

Remarks

This is Hironaka's theorem. The morphism $\pi$ can be chosen as a composition of blow-ups along nonsingular centres that lie over the singular points of $X$. The proof is an induction on dimension: one restricts to a hypersurface of maximal contact and uses an invariant of the singularity that drops after each blow-up. For curves, in any characteristic, the normalization is a resolution. Example: the cusp $y^2 = x^3$ is resolved by $\mathbb{A}^1 \to X$, $t \mapsto (t^2, t^3)$, which is bijective and an isomorphism away from $t = 0$. In characteristic $p > 0$ the statement is known in dimension at most $3$, and it is open in general.

Going deeper

For a student

Resolution lets one prove results for smooth varieties and then transfer them to singular ones. It is used to define invariants of singularities, such as rational singularities, and to compactify smooth varieties by a divisor with simple normal crossings. Resolution by blow-ups is also a basic step of the minimal model program, which starts from a smooth model of a birational class.