Editing Algebraic surface (section)

{{Short description|Zero set of a polynomial in three variables}}
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{{technical|date=May 2023}}
{{confusing|reason=it is unclear when the article is restricted to smooth algebraic surfaces over the complexes, and when more general cases are considered|date=May 2023}}}}
In [[mathematics]], an '''algebraic surface''' is an [[algebraic variety]] of [[dimension of an algebraic variety|dimension]] two. In the case of geometry over the field of [[complex number]]s, an algebraic surface has complex dimension two (as a [[complex manifold]], when it is [[Algebraic curve#Singularities|non-singular]]) and so of dimension four as a [[smooth manifold]].

The theory of algebraic surfaces is much more complicated than that of [[algebraic curve]]s (including the [[compact space|compact]] [[Riemann surface]]s, which are genuine [[Surface (topology)|surfaces]] of (real) dimension two).  Many results were obtained, but, in the [[Italian school of algebraic geometry ]], and are up to 100 years old.