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An algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
2
votes
Göttsche's formula for non-compact complex surfaces?
Yes. This is Theorem 5.2.1 of The Douady space of a complex surface by de Cataldo and Migliorini (Math.Ann.)
2
votes
Embedding degree 1 Del Pezzo surfaces in $\mathbb{P}(1,1,2,3)$
Me and my graduate student have a method, which this margin is too narrow to contain. Please get in touch if this is of further interest.