---
tag: KFTECAATKG
kind: concept
name: Algebraic surface
msc: [14J10, 14J25]
specializes: [HV6YBS0YDH]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_surface
same_as:
  wikidata: Q1434805
  wikipedia: Algebraic surface
---

## Summary

An algebraic surface is an [[algebraic variety|HV6YBS0YDH]] of [[dimension|36X2WTHYCB]] two.
Over the complex numbers a smooth projective algebraic surface is a compact complex manifold of complex dimension two, so of real dimension four.
Surfaces are the first case where birational geometry is subtle, and their classification by Enriques and Kodaira is a model for higher dimensions.

## Explanations

### L1 human

A curve is described by one equation in two variables, and a surface by one equation in three variables, such as $x^2 + y^2 + z^2 = 1$.
Locally a surface looks like a piece of the plane, so it has two directions of freedom.
Over the complex numbers each of these directions counts twice, so a complex surface is a four-dimensional real object.
Examples are the projective plane, the product of two curves, and the zero set of a polynomial in projective three-space.