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 formula_1 and the geometric genus formula_2 turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the irregularity for the classification of them. 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):   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). One of the fundamental theorems for the birational geometry of surfaces is Castelnuovo's theorem. This states that any birational map between algebraic surfaces is given by a finite sequence of blowups and blowdowns. The Nakai criterion says that:  Ample divisors have a nice property such as it is the pullback of some hyperplane bundle of projective space, whose properties are very well known. Let formula_3 be the abelian group consisting of all the divisors on "S". Then due to the intersection theorem is viewed as a quadratic form. Let then formula_6 becomes to be a numerical equivalent class group of "S" and also becomes to be a quadratic form on formula_8, where formula_9 is the image of a divisor "D" on "S".