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date: 2022-12-29 21:17
modification date: Thursday 29th December 2022 21:17:17
title: "Hironaka's resolution of singularities"
aliases: [Hironaka's resolution of singularities]
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# Hironaka's resolution of singularities



Theorem 19.1 (Hironaka). (a) Given a projective variety $Y$, there exists a smooth projective variety $Y^{\prime}$ and a birational morphism $\varphi: Y^{\prime} \rightarrow Y$ (called a desingularization of $Y$ ) such that $\varphi$ is an isomorphism over the smooth locus $\operatorname{reg} Y$, that is,
$$\left.\varphi\right|_{\varphi^{-1}(\operatorname{reg} Y)}: \varphi^{-1}(\operatorname{reg} Y) \stackrel{\cong}{\longrightarrow} \operatorname{reg} Y$$
(b) Given a smooth projective variety $X$ and a projective subvariety $Y \subset X$, there exists a smooth projective variety $W$ and a birational morphism $\varphi: W \rightarrow X$ such that $\varphi^{-1}(Y) \subset W$ is a divisor with simple normal crossings.
(c) If $Y \subset X$ as in (b) is a hypersurface, then there is a unique irreducible component $Y^{\prime}$ of the exceptional set $\varphi^{-1}(Y)$ which dominates $Y$. The restriction $\left.\varphi\right|_{Y^{\prime}}: Y^{\prime} \rightarrow Y$ is a desingularization satisfying (5) (in this case $\varphi$ is called an embedded desingularization, or embedded resolution of $Y$ ).
(d) The morphism $\varphi$ in (a) and (b) can be chosen to be a composition of blowups with smooth reduced centers.

## Surfaces

In the case of surfaces, we have the following facts.
Theorem 19.2 (Zariski). Any projective surface can be desingularized via a sequence of transforms, which repeatedly alternates normalizations and blowups of maximal ideals. The resulting smooth surface is again projective.

Theorem 19.3 (Minimal resolution of surface singularities). Any normal projective surface $Y$ admits a unique minimal resolution of singularities, that is, a birational morphism $\varphi: X \rightarrow Y$ such that
1) $X$ is a smooth projective surface;
2) $\varphi$ is an isomorphism over the regular locus $\operatorname{reg} Y$;
3) any other desingularization $\varphi^{\prime}: X^{\prime} \rightarrow Y$ satisfying 1) and 2) admits a factorization $\varphi^{\prime}=\sigma \circ \varphi$, where $\sigma: X^{\prime} \rightarrow X$ is a composition of blowups of points over the exceptional locus of $\varphi$.