---
tag: EQSHJPE7H8
kind: concept
name: Quasi-projective variety
msc: [14A10, 14N05]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-projective_variety
same_as:
  wikidata: Q7269519
  wikipedia: Quasi-projective variety
---

## Summary

A quasi-projective variety is an open subset of a projective variety, with the induced Zariski topology.
The class contains both the affine varieties and the projective varieties, and it is closed under taking open and closed subvarieties.
In classical algebraic geometry, the quasi-projective varieties are exactly the varieties one works with.

## Explanations

### L1 human

Start with a projective variety and remove a closed piece of it, defined by some equations.
What is left is a quasi-projective variety.
Removing the line at infinity from the projective plane leaves the affine plane, so affine varieties are quasi-projective.
Removing a single point from the affine plane leaves a quasi-projective variety that is neither affine nor projective.