---
tag: A4H4CZ2V35
kind: concept
name: Projective variety
msc: [14N05, 14A10]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_variety
same_as:
  wikidata: Q3554818
  wikipedia: Projective variety
---

## Summary

A projective variety is an irreducible subset of projective space $\mathbb{P}^n$ over an algebraically closed field cut out by homogeneous polynomial equations.
Projective varieties are the algebraic analogue of compact spaces.
Over $\C$ they are compact in the usual topology, and images of projective varieties under morphisms are closed.
Most classical algebraic geometry, such as plane curves, Bézout's theorem and intersection theory, takes place on projective varieties.

## Explanations

### L1 human

In projective space a point has homogeneous coordinates that are only defined up to scaling.
So an equation makes sense on points only if scaling does not change whether it holds, which is true for homogeneous polynomials such as $x^2 + y^2 - z^2$.
The zero set of $x^2 + y^2 - z^2$ in the projective plane is a conic.
In the chart $z = 1$ it is the circle $x^2 + y^2 = 1$, and the projective version adds no points at infinity over $\R$ but two over $\C$.