Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, the abstract algebra, in particular the study of commutative rings, linked to the geometry. They can be briefly defined as the study of the zeros structures describe algebraic equations.

Geometric structures as a set of zeros

In algebraic geometry geometrical structures are defined as a set of zeros of a set of polynomials. For example, can be the two-dimensional sphere in three-dimensional Euclidean space R 3 as the set of all points (x, y, z) defining, in which:

A " tilted " circle in R3 can be defined as the set of all points (x, y, z) that satisfy the following two Polynombedingungen:

Affine varieties

Is generally K be a field and S be a set of polynomials in n variables with coefficients in K, then V (S) is defined as the subset of Kn, which consists of the common zeros of the polynomials in S. A lot of this form is called an affine variety. The affine varieties defining a topology to Kn, called Zariski topology. As a consequence of the Hilbert basis set between each variety can be defined by a finite number of polynomial equations. One variety is called irreducible if it is not the union of two proper closed subsets. It turns out that a variety is irreducible if and only if the polynomials that define them generate a prime ideal of the polynomial ring. The correspondence between varieties and ideals is a central theme of algebraic geometry. The simple fact is a dictionary between geometric concepts such as variety, irreducible, and so on algebraic concepts, such as ideal, a prime ideal and so specify.

At any variety V can associate a commutative ring, called ring coordinates. It consists of all the polynomial defined on the variety. The prime ideals of this ring are in correspondence with the irreducible subvarieties of V; if K is algebraically closed, what is usually assumed, then correspond to the points of V the maximal ideals of the coordinate ring ( Hilbert Nullstellensatz ).

Projective space

Instead of working in the affine space Kn, one typically goes through to the projective space. The main advantage of this is that the number of intersection points of two varieties can then be easily determined with the help of the theorem of Bézout.

In the modern view of the correspondence between varieties and their coordinate rings is reversed: you go from an arbitrary commutative ring and defines an associated variety using its prime ideals. From the prime ideals a topological space is initially designed, the spectrum of the ring. In the most general formulation, this leads to Alexander Grothendieck's schemes.

An important class of varieties are the abelian varieties. These are projective varieties whose points form an abelian group. The typical examples are elliptic curves, which play an important role in the proof of Fermat's Last theorem. Another important application is the elliptic curve cryptography.

Algorithmic calculations

While primarily abstract statements about the structure of varieties have been taken in algebraic geometry for a long time, algorithmic techniques have recently been developed that allow the efficient computation with Polynomidealen. The main tool is the Gröbner bases that are implemented in most modern computer algebra systems.

Historical Overview

Algebraic geometry has been developed in many parts of the Italian geometers of the early twentieth century. Her work has been profound, but not on a sufficiently rigorous basis. The commutative algebra ( as the study of commutative rings and their ideals ) was developed by David Hilbert, Emmy Noether and also developed other at the beginning of the twentieth century. They already had the geometric applications in mind. In the 1930s and 1940s André Weil noted that the algebraic geometry had to be put on a strict basis and developed a corresponding theory. In the 1950s and 1960s revised Jean-Pierre Serre and Alexander Grothendieck especially those basics with the use of sheaves and later with the use of the schemes.

Examples of affine varieties

• Conic circle
• Ellipse
• Parabola
• Hyperbola

• Cartesian sheet
• Cisoids
• Strophoid

• Conchoid
• Pascal's snail