Algebraic surface

In algebraic geometry, a surface in other ways than in the differential geometry and topology is examined. An algebraic surface is defined by means of polynomials that are mathematically well recorded. Using the tools of abstract algebra symmetries and singularities are detected only by examination of the polynomials and their solution sets. The variety and the abundance of theory is much larger than that of algebraic curves in algebraic surfaces.

Definition

An algebraic surface is always an algebraic variety, it is thus described by polynomial equations. The points belonging to the surface are exactly the solutions of the equations. Since there is a notion of dimension for algebraic varieties, can be surfaces, ie varieties of dimension two, differ from curves or higher-dimensional varieties.

A complex variety that does not have singularities at the same time a complex manifold. In this case, important to distinguish between complex and real dimension. For a Riemann surface is a complex and real two-dimensional, a Danielewski surface is a complex two-dimensional and thus real four-dimensional. A Riemann surface is therefore not a complex area, but a complex curve.

Examples

Examples of algebraic surfaces is obtained as follows:

Let k be an algebraically closed field, for example, the field of complex numbers, and a non-constant polynomial in three variables x, y and z with coefficients in k. Then the zero set of f, ie, the set is an affine algebraic surface.

The simplest surfaces are given by planes which are defined by linear equations, where A, B and C are not all zero. Another example is the spherical surface, which is defined by the equation.