---
tag: E4F316AR19
kind: definition
name: Blowing up
msc: ["14E05", "14E15"]
defines: C4KGYAFPN7
uses: [5A35TYZD02, N3PGYJJQ7W, A39XQHCVNW, FHV05ZK99F, ZT9A6JK8QF]
sources:
  - url: https://en.wikipedia.org/wiki/Blowing_up
---

## Definition

Let $X$ be a Noetherian [[scheme|5A35TYZD02]] and let $Z \subseteq X$ be a closed subscheme with [[ideal sheaf|N3PGYJJQ7W]] $\mathcal{I} \subseteq \mathcal{O}_X$.
The *blow-up* of $X$ along $Z$ is the scheme
$$\operatorname{Bl}_Z X = \operatorname{Proj}_X \bigoplus_{d \ge 0} \mathcal{I}^d,$$
given by the relative [[Proj construction|A39XQHCVNW]], with $\mathcal{I}^0 = \mathcal{O}_X$, together with its projection $\pi \colon \operatorname{Bl}_Z X \to X$.
The closed subscheme $E = \pi^{-1}(Z)$ is the *exceptional divisor*, and $Z$ is the *centre* of the blow-up.

## Remarks

The ideal sheaf $\pi^{-1}\mathcal{I} \cdot \mathcal{O}_{\operatorname{Bl}_Z X}$ of $E$ is invertible, so $E$ is an effective [[Cartier divisor|FHV05ZK99F]].
The morphism $\pi$ is projective, and it is an isomorphism over $X \setminus Z$.
It has a universal property: every morphism $f \colon Y \to X$ such that $f^{-1}\mathcal{I} \cdot \mathcal{O}_Y$ is invertible factors uniquely through $\pi$.

Example: the blow-up of $\mathbb{A}^2 = \operatorname{Spec} k[x, y]$ at the origin, with $\mathcal{I} = (x, y)$, is the closed subvariety
$$\set{((x, y), [u : v]) \in \mathbb{A}^2 \times \mathbb{P}^1 : x v = y u}.$$
Over a point $(x, y) \ne (0, 0)$ the condition forces $[u : v] = [x : y]$, so $\pi$ is bijective there.
Over the origin every $[u : v]$ is allowed, so $E \cong \mathbb{P}^1$, the [[projective line|ZT9A6JK8QF]] of directions through the origin.

Non-example: the projection $\mathbb{A}^2 \times \mathbb{P}^1 \to \mathbb{A}^2$ is not the blow-up at the origin.
It fails to be an isomorphism over $\mathbb{A}^2 \setminus \set{0}$, since every fibre is a whole $\mathbb{P}^1$.

Blowing up an effective Cartier divisor changes nothing: $\pi$ is then an isomorphism.