---
tag: YWB981MEF2
kind: concept
name: Toric variety
msc: [14M25]
sources:
  - url: https://en.wikipedia.org/wiki/Toric_variety
same_as:
  wikidata: Q1528306
  wikipedia: Toric variety
---

## Summary

A toric variety is a variety that contains an [[algebraic torus|66FQFAJNS4]] as a dense open subset.
The action of the torus on itself must extend to the whole variety.
Normal toric varieties correspond to combinatorial objects called fans, built from rational polyhedral cones.
So geometric questions about them turn into questions about cones and polytopes, such as [[Newton polytopes|CYTTX8KHC8]].
They are a standard testing ground for conjectures and constructions in algebraic geometry.

## Explanations

### L1 human

Start with the points $(x, y)$ where both coordinates are nonzero; this is a two-dimensional torus.
Adding back the coordinate axes gives the plane, and adding a line at infinity gives the projective plane.
In each case the torus sits inside as a dense open part, and scaling the coordinates still makes sense on the whole space.
Such spaces are toric varieties, and the way they are glued from the torus is recorded by a picture of cones in a lattice.