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. Enriques–Kodaira classification In the case of dimension one, varieties are classified by only the topological genus, but, in dimension two, one needs to distinguish the arithmetic genus and the geometric genus because one cannot distinguish birationally only the topological genus. Then, irregularity is introduced for the classification of varieties. A summary of the results (in detail, for each kind of surface refers to each redirection), follows: Examples of algebraic surfaces include (κ is the Kodaira dimension): κ = −∞: the projective plane, quadrics in P3, cubic surfaces, Veronese surface, del Pezzo surfaces, ruled surfaces κ = 0 : K3 surfaces, abelian surfaces, Enriques surfaces, hyperelliptic surfaces κ = 1: elliptic surfaces κ = 2: surfaces of general type. For more examples see the list of algebraic surfaces. The first five examples are in fact birationally equivalent. That is, for example, a cubic surface has a function field isomorphic to that of the projective plane, being the rational functions in two indeterminates. The Cartesian product of two curves also provides examples. The birational geometry of algebraic surfaces is rich, because of blowing up (also known as a monoidal transformation), under which a point is replaced by the curve of all limiting tangent directions coming into it (a projective line). Certain curves may also be blown down, but there is a restriction (self-intersection number must be −1).