Algebraic surface

In algebraic geometry , a surface is examined in a different way than in differential geometry and topology . An algebraic surface is defined by means of polynomials that are mathematically well recorded. With the tools of abstract algebra , symmetries and singularities are recognized solely by examining the polynomials and their solution sets . The variety of shapes and the wealth of theory is much greater with algebraic surfaces than with algebraic curves .

definition

An algebraic surface is always an algebraic variety , so it is described by polynomial equations. The points belonging to the area are exactly the solutions to the equations. Since there is a dimension concept for algebraic varieties , surfaces, i.e. varieties of dimension two, can be distinguished from curves or higher-dimensional varieties.

A complex variety that has no singularities is also a complex manifold . In this case a distinction must be made between complex and real dimensions. A Riemann surface is complex one-dimensional and real two-dimensional, a Danielewski surface is complex two-dimensional and therefore real four-dimensional. A Riemann surface is therefore not a complex surface, but a complex curve.

Examples

Examples of algebraic surfaces are 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 set of zeros of f , that is, the set is an affine algebraic surface. ${\displaystyle f}$ ${\displaystyle \{(x,y,z)\in k^{3}|f(x,y,z)=0\}}$

The simplest areas are given by planes 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 . ${\displaystyle ax+by+cz+d=0}$ ${\displaystyle x^{2}+y^{2}+z^{2}-1=0}$

See also

• Quadric
• Curve (algebraic geometry)

Web links

• Some nice algebraic surfaces by Gerhard Brunthaler.