---
tag: C4KGYAFPN7
kind: concept
name: Blowing up
msc: ["14E05", "14E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Blowing_up
same_as:
  wikidata: Q1036595
  wikipedia: Blowing up
---

## Summary

Blowing up is a construction that replaces a closed subvariety of a variety by the set of normal directions to it.
The result maps birationally onto the original variety and is an isomorphism away from the centre.
The preimage of the centre, the exceptional divisor, is a hypersurface.
Blowing up is the basic tool of [[birational geometry|51YVV85GWB]] and of [[resolution of singularities|MB52NDA6WN]].

## Explanations

### L1 human

To blow up the origin of the plane, keep every other point, but replace the origin by a whole projective line, one point for each line through the origin.
Lines through the origin that used to meet there are now separated, because they arrive at different points of the new line.
A picture is a spiral staircase over the plane: above the origin sits a full circle of directions, and above other points one point.
For example, the curve $y^2 = x^2(x + 1)$ crosses itself at the origin, and after blowing up its two branches pass through different points and no longer meet.