Algebraic surface

In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the complex number field, an algebraic surface is therefore of complex dimension two (as a complex manifold) 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.

Examples of algebraic surfaces include:

• the projective plane
• quadrics in P³
• cubic surfaces
• Del Pezzo surfaces
• ruled surfaces
• K3 surfaces
• abelian surfaces
• surfaces of general type.

The first three 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 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).

Basic results on algebraic surfaces include the Hodge index theorem, and the division into five groups of birational equivalence classes called the classification of algebraic surfaces. The general type class, of Kodaira dimension 2, is very large (degree 5 or larger for a non-singular surface in P³ lies in it, for example).

There are essential three Hodge number invariants of a surface. Of those, h^1,0 was classically called the irregularity and denoted by q; and h^2,0 was called the geometric genus p_g. The third, h^1,1, is not a birational invariant.

The Riemann-Roch theorem for surfaces was first formulated by Max Noether. The families of curves on surfaces can be classified, in a sense, and give rise to much of their interesting geometry.