Algebraic geometry

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

Zeroes of simultaneous polynomials

In algebraic geometry, the geometric objects studied are defined as the set of zeroes of a number of polynomials: meaning the set of common zeroes, or equally the set defined by one or several simultaneous polynomial equations. 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.

x² + y² + z² -1 = 0

x + y + z = 0

Affine varieties

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; it carries a natural topology, the Zariski topology, the closed sets of which are also defined by polynomial equations. As a consequence of Hilbert's basis theorem, every variety can be defined by 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.

Coordinate ring of a variety

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).

Projective theory

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 Bézout's theorem.

Background to the current point of view on the subject

In the modern view, the correspondence between variety and coordinate ring is turned around: one starts with an abstract commutative ring and defines a corresponding variety via its prime ideals. The prime ideals are first turned into a topological space, 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 which are varieties whose points form an abelian group. The prototypical examples are the elliptic curves which were instrumental in the proof of Fermat's last theorem and are also used in elliptic 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 1930s and 1940s Oscar Zariski[?], Andre Weil and others realized that a requirement existed for an axiomatic algebraic geometry on a rigorous basis. For a while there were several foundational theories used. In the 1950s and 1960s Jean-Pierre Serre and Grothendieck recast the foundations making use of the theory of sheaves. Later, from about 1960, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilised in the 1970s, and applications were made, both to number theory and to more classical geometric questions on algebraic varieties, singularities and moduli.

External links: Kevin R. Coombes: Algebraic Geometry: A Total Hypertext Online System, http://odin.mdacc.tmc.edu/~krc/agathos/. Graduate level online textbook for students familiar with abstract algebra.