---
tag: 43DJVJYRBN
kind: concept
name: Complete intersection
msc: [14M10, 13C40]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_intersection
same_as:
  wikidata: Q5156502
  wikipedia: Complete intersection
---

## Summary

A complete intersection is a variety in affine or projective space whose ideal can be generated by as many equations as its codimension.
Each equation cuts the dimension down by exactly one, so the variety is defined by the smallest possible number of equations.
Complete intersections are among the simplest varieties to study.
Their degree is the product of the degrees of the equations, and their coordinate rings have good algebraic properties.

## Explanations

### L1 human

In three-dimensional space, one equation usually defines a surface and two equations usually define a curve.
A curve in space is a complete intersection when two equations are enough to describe it exactly.
A line, such as $x = y = 0$, is a complete intersection.
The twisted cubic curve, the image of $t \mapsto (t, t^2, t^3)$ completed in projective space, is a curve in $\mathbb{P}^3$ whose ideal needs three equations, so it is not a complete intersection.