Algebraic Geometry

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The algebraic geometry is a branch of mathematics that the abstract algebra , especially the study of commutative rings , with the geometry linked. It can be briefly described as the study of the zeros of algebraic equations .

Geometric structures as a set of zeros

Sphere and the "tilted" circle

In algebraic geometry, geometric structures are defined as a set of zeros in a set of polynomials . For example, the two-dimensional unit sphere in three-dimensional Euclidean space can be defined as the set of all points for which:


A "tilted" circle in can be defined as the set of all points that meet the following two polynomial conditions:

Affine varieties

If in general there is a field and a set of polynomials in variables with coefficients in , then the set of zeros is defined as that subset of which consists of the common zeros of the polynomials in . Such a set of zeros is called an affine variety . The affine varieties define a topology on , called the Zariski topology . As a consequence of Hilbert's basis theorem , each variety can be defined by only finitely many polynomial equations. A variety is called irreducible if it is not the union of two true closed subsets. It turns out that a variety is irreducible if and only if the polynomials that define it generate a prime ideal of the polynomial ring. The correspondence between varieties and ideals is a central theme of algebraic geometry. One can almost give a dictionary between geometric terms, such as variety, irreducible, etc., and algebraic terms, such as ideal, prime ideal, etc.

Each variety can be associated with a commutative ring, the so-called coordinate ring . It consists of all of the polynomial functions defined on the variety. The prime ideals of this ring correspond to the irreducible sub-varieties of ; if it is algebraically closed , which is usually assumed, then the points correspond to the maximum ideal of the coordinate ring ( Hilbert's zero theorem ).

Projective space

Instead of working in affine space , one typically moves to projective space . The main advantage is that the number of intersections of two varieties can then easily be determined with the help of Bézout's theorem .

In the modern view, the correspondence between varieties and their coordinate rings is reversed: one starts with any commutative ring and defines an associated variety using its prime ideals. First a topological space is constructed from the prime ideals , the spectrum of the ring. In its most general formulation, this leads to Alexander Grothendieck's schemes .

An important class of varieties are the Abelian varieties . These are projective varieties , the points of which form an Abelian group . The typical examples of this are elliptic curves , which play an important role in the proof of Fermat's Great Theorem . Another important area of ​​application is cryptography with elliptic curves.

Algorithmic calculations

While in algebraic geometry mainly abstract statements about the structure of varieties have been made for a long time, algorithmic techniques have recently been developed that allow efficient calculation with polynomial ideals. The most important tools are the Gröbner bases , which are implemented in most of today's computer algebra systems.

historical overview

Algebraic geometry was largely developed by the Italian geometricians of the early twentieth century. Their work was in-depth 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 also developed and other at the beginning of the twentieth century. They already had the geometric applications in mind. In the 1930s and 1940s André Weil realized that algebraic geometry had to be placed on a strict basis, and developed a corresponding theory. In the 1950s and 1960s, Jean-Pierre Serre, and especially Alexander Grothendieck, revised these principles using Sheaves and later using the schemes . Today there are many very different sub-areas of algebraic geometry, on the one hand abstract theory in the footsteps of Grothendieck, on the other hand areas in which combinatorics and discrete mathematics are used, such as toric geometry or tropical geometry .

Examples of affine varieties


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