Algebraic geometry Surfaces and higher-dimensional varieties Concept
Algebraic surface
Summary
An algebraic surface is an algebraic variety of dimension 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.
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| Tag | KFTECAATKG |
|---|---|
| URI | m2://KFTECAATKG |
| Kind | concept |
| Writing | W0 stub |
| MSC | 14J10, 14J25 |
| Specializes | Algebraic variety |
| Same as | |
| Changed | 2 November 2026 |
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Used by
- Enriques–Kodaira classification
- Elliptic surface
- Enriques surface
- Surface of general type
- Del Pezzo surface
- Rational surface
- K3 surface
- Castelnuovo's rationality criterion
- Bogomolov–Miyaoka–Yau inequality
- Castelnuovo's contraction theorem
- Hodge index theorem
- Ruled surface