---
tag: 51YVV85GWB
kind: concept
name: Birational geometry
msc: ["14E05", "14E30", "14E08"]
sources:
  - url: https://en.wikipedia.org/wiki/Birational_geometry
same_as:
  wikidata: Q1879333
  wikipedia: Birational geometry
---

## Summary

Birational geometry is the part of algebraic geometry that studies varieties up to [[birational equivalence|KA5890R9YM]].
Two varieties are birationally equivalent when they become isomorphic after removing lower-dimensional closed subsets, or equivalently when they have isomorphic [[function fields|2D5SS6WM14]].
Its main questions are which varieties are [[rational|XV4K9GREJ9]], and how to choose a simplest variety in each birational class.
The [[minimal model program|Q8CTDCZH01]] is its central project in higher dimensions.

## Explanations

### L1 human

In birational geometry one is allowed to cut out a curve or a point and glue in something else, as long as the bulk of the variety stays the same.
The plane $\mathbb{P}^2$ and the quadric surface $\mathbb{P}^1 \times \mathbb{P}^1$ are not isomorphic, but they agree on dense open subsets, so they are birational.
[[Blowing up|C4KGYAFPN7]] a point is the basic move: it replaces a point by a line of directions.
Numbers that do not change under such moves, such as the geometric genus or the Kodaira dimension, are the main tools to tell birational classes apart.