Algebraic surface

In mathematics, an algebraic surface is an algebraic variety of dimension two. Thus, an algebraic surface is a solution of a set of polynomial equations, in which there are two independent directions at every point. An example of an algebraic surface is the sphere, which is determined by the single polynomial equation

x

2

+

y

2

+

z

2

=

1.

{\displaystyle x^{2}+y^{2}+z^{2}=1.}

Studying the intrinsic geometry of algebraic surfaces is a central topic in algebraic geometry. The theory is much more complicated than for algebraic curves (one-dimensional cases), and was developed substantially by the Italian school of algebraic geometry in the late 19th and early 20th centuries.

It remains an active field of research.

In the simplest cases, algebraic surfaces are studied as algebraic varieties over the complex numbers. For example, the familiar sphere (for real

x

,

y

,

z

{\displaystyle x,y,z}

), becomes a complex (affine) quadric surface, which simultaneously incorporates the sphere and hyperboloids of one and two sheets, and this allows some complications (such as the topology: whether the surface is connected, or simply connected) to be deferred somewhat. Higher degree surfaces include, for example, the Kummer surface. The classification of algebraic surfaces is much more intricate than the classification of algebraic curves, which have dimension one, and is already quite complicated.