Algebraic surface

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.