Algebraic surface

In mathematics, 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.

The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry, and are up to 100 years old.

In the case of dimension one varieties are classified by only the topological genus, but dimension two, the difference between the arithmetic genus ${\displaystyle p_{a}}$ and the geometric genus ${\displaystyle p_{g}}$ turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the irregularity for the classification of them. Let's summarize the results. (in detail, for each kind of surfaces refer to each redirections)

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