Algebraic geometry

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Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially the study of commutative rings, and geometry. It can be seen as the study of solutions sets of systems of algebraic equations.

In algebraic geometry, geometric structures are defined as the set of zeros of a number of polynomials. For instance, the two-dimensional sphere in three-dimensional Euclidean space R3 could be defined as the set of all points (x, y, z) with

x2 + y2 + z2 -1 = 0.

A "slanted" circle in R3 can be defined as the set of all points (x, y, z) which satisfy the two polynomial equations

x2 + y2 + z2 -1 = 0
x + y + z = 0

In general, if F is a field and S a set of polynomials over F in n variables, then V(S) is defined to be the subset of Fn which consists of the simultaneous zeros of the polynomials in S. A set of this form is called an affine variety, and it carries a natural topology, the Zariski topology which is also defined by polynomial equations. As a consequence of Hilbert's basis theorem, every variety can be defined by only finitely many polynomial equations. A variety is called irreducible if it cannot be written as the union of two smaller varieties. It turns out that a variety is irreducible if and only if the polynomials defining it generate a prime ideal of the polynomial ring. This correspondence of irreducible varieties and prime ideals is a central theme of algebraic geometry.

To every variety V one can associate a commutative ring, the coordinate ring, consisting of all polynomial functions defined on the variety. The prime ideals in this ring correspond to the irreducible subvarieties of V; if F is algebraically closed, which is usually assumed, then the points of V correspond to the maximal ideals of the coordinate ring (Hilbert's Nullstellensatz).

Instead of working in the affine space Fn, one typically employs projective space, the main advantage being that the number of intersection points between varieties can then be easily calculated using Bezout's theorem.

In the modern view, the correspondence between variety and coordinate ring is turned around: one starts with an abstract commutative ring and defines a correspoinding variety via its prime ideals. In the most general formulation, this leads to Alexander Grothendieck's schemes.

An important class of varieties are the abelian varieties which are varieties whose points form an abelian group. The prototypical examples are the elliptical curves which were instrumental in the proof of Fermat's last theorem and are also used in elliptical curve cryptography.

While much of algebraic geometry is concerned with abstract and general statements about varieties, methods for the effective computation with concretely given polynomials have also been developed. The most important is the technique of Grobner bases which is employed in all computer algebra systems.

Algebraic geometry was developed largely by the Italian geometers in the early part of the 20-th century. Their work was deep but not on a sufficiently rigorous basis. Commutative algebra (as the study of commutative rings and their ideals) was developed by David Hilbert, Emmy Noether and others, also in the 20-th century, with the geometric applications in mind. In the 1930's and 1940's Andre Weil realized that putting algebraic geometry on a rigorous basis was needed and he gave such a theory. In the 1950's and 1960's Jean-Pierre Serre and particularly Grothendieck recast the foundations making use of the theory of sheaves and, later, schemes.