---
tag: 9G91CJJ0JQ
kind: definition
name: Algebraic surface
msc: [14J10, 14J25]
defines: KFTECAATKG
uses: [HV6YBS0YDH, 36X2WTHYCB, 0PF6TM8TWT]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_surface
---

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]].
An *algebraic surface* over $k$ is an [[algebraic variety|HV6YBS0YDH]] over $k$ of [[dimension|36X2WTHYCB]] $2$.

## Remarks

In the classification theory, "surface" usually means a smooth projective surface, and these extra conditions are then stated.
Example: the projective plane $\mathbb{P}^2$ and the product $\mathbb{P}^1 \times \mathbb{P}^1$ are algebraic surfaces.
Example: the zero set in $\mathbb{P}^3$ of one nonconstant irreducible homogeneous polynomial is an algebraic surface, since a hypersurface in $\mathbb{P}^3$ has dimension $3 - 1 = 2$.
Non-example: a plane curve such as $y^2 = x^3 - x$ in $\mathbb{A}^2$ is an algebraic variety of dimension $1$, not $2$.
Non-example: the affine space $\mathbb{A}^3$ is a variety of dimension $3$.
Over $\C$, by Chow's theorem, a smooth projective algebraic surface is the same as a compact complex surface that embeds in projective space.
Not every compact complex surface is of this kind.