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Examples of algebraic surfaces include:

• the projective plane
• quadrics in P³
• cubic surfaces
• Del Pezzo surfaces
• ruled surfaceIn geometry, a surface is ruled if through every point of there is a straight line that lies on. The plane, cylinder, and cone are the most familiar examples. The first two are special cases of ruled quadrics (which also include the hyperbolic paraboloid,s
• K3 surfaceA K3 manifold is a hyperkahler manifold of real dimension 4, i. a four-dimensional connected manifold with SU(2) holonomy. All K3 manifolds can be showed to be diffeomorphic to each other. These manifolds first arose in algebraic geometry: they are imports
• 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 fieldIn algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. The ring of regular functions on a variety V defined over a field K is an integral domain if and only if the variety isomorphic to that of the projective plane, being the rational functionIn mathematics, a rational function is a ratio of polynomials. For a single variable x a typical rational function is therefore P ''x Q ''x where P and Q are polynomials in x as indeterminate, and Q isn't the zero polynomial. Any non-zero polynomial Q iss in two indeterminates. The birational geometryGeometry In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension 2, the birational geometry of algebrai 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 lineIn mathematics, the projective line is a fundamental example of an algebraic curve. It may be defined over any field K as the set of one-dimensional subspaces of the two-dimensional vector space K''2; it does carry other geometric stuctures. It may be den). 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 surface s. 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.