---
date: 2022-12-29 21:16
modification date: Thursday 29th December 2022 21:16:35
title: "Castelnuovo"
aliases: [Castelnuovo]
---

Last modified date: NaN

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- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Hironaka's resolution of singularities](Hironaka's%20resolution%20of%20singularities)

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# Castelnuovo


Theorem 18.3 (Castelnuovo). Let $X$ be a smooth (abstract) algebraic surface, and let $E$ be a smooth projective curve in $X$. There exists a contraction $\sigma: X \rightarrow Y$ with exceptional curve $E$, where $Y$ is a smooth algebraic surface, if and only if $E$ is a $(-1)$-curve, that is, $E \cong \mathbb{P}^{1}$ and $E^{2}=-1$.


Theorem $18.7$ (Decomposition of birational maps of surfaces). Let $X$ and $Y$ be smooth projective surfaces.
(a) Any birational morphism $f: X \rightarrow Y$ can be decomposed into a sequence of blowdowns of (-1)-curves.
(b) Any birational map $F: X \rightarrow Y$ can be decomposed into a sequence of blowups of points followed by a sequence of blowdowns of (-1)-curves. The first sequence yields the birational morphism $g: W \rightarrow X$ in diagram (4), and the second the birational morphism $f: W \rightarrow Y$ in (4).