---
tag: XJM2MT4S3Y
kind: concept
name: Algebraic curve
msc: [14H99, 14H50]
specializes: [HV6YBS0YDH]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_curve
same_as:
  wikidata: Q266237
  wikipedia: Algebraic curve
---

## Summary

An algebraic curve is an [[algebraic variety|HV6YBS0YDH]] of dimension one, such as the set of solutions of one polynomial equation in two unknowns.
Curves are the first case of algebraic geometry beyond points, and their theory is the most complete.
Their main invariant is the [[genus|6W8TDV9VVN]].
Over $\C$, smooth projective curves are the same as compact [[Riemann surfaces|5K0N8QMV6Q]].

## Explanations

### L1 human

A curve is a geometric object with one degree of freedom, cut out by polynomial equations.
The circle $x^2 + y^2 = 1$ and the cubic $y^2 = x^3 - x$ are curves in the plane.
The twisted cubic $\set{(t, t^2, t^3)}$ is a curve in space that does not lie in any plane.
Over the complex numbers each curve is a real surface, and a smooth projective connected curve looks like a sphere with some number of handles; that number is its genus.