---
tag: HV6YBS0YDH
kind: concept
name: Algebraic variety
msc: [14A10]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_variety
same_as:
  wikidata: Q648995
  wikipedia: Algebraic variety
---

## Summary

An algebraic variety is the central object of algebraic geometry.
It is a geometric space that is locally the set of solutions of a system of polynomial equations.
In the classical setting over an algebraically closed field, the varieties are the affine, quasi-affine, projective and quasi-projective varieties.
In modern language, a variety is an integral separated scheme of finite type over a field.
This allows varieties that do not sit inside any projective space.
Conventions differ between books, mainly on whether a variety must be irreducible.

## Explanations

### L1 human

A variety is a shape described by polynomial equations, like a line, a circle, a sphere or the curve $y^2 = x^3 + x$.
Locally it always looks like the solution set of equations in some $k^n$, and it may be completed by points at infinity in projective space.
The word usually means a single piece: the union of two crossing lines is an algebraic set with two varieties as its pieces.
Varieties are studied with algebra, through the rings of polynomial functions that live on them.