Algebraic geometry

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.
The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular points, inflection points and points at infinity. More advanced questions involve the topology of the curve and the relationship between curves defined by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. As a study of systems of polynomial equations in several variables, the subject of algebraic geometry begins with finding specific solutions via equation solving, and then proceeds to understand the intrinsic properties of the totality of solutions of a system of equations. This understanding requires both conceptual theory and computational technique.
In the 20th century, algebraic geometry split into several subareas.

The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.
Real algebraic geometry is the study of the real algebraic varieties.
Diophantine geometry and, more generally, arithmetic geometry is the study of algebraic varieties over fields that are not algebraically closed and, specifically, over fields of interest in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, and p-adic fields.
A large part of singularity theory is devoted to the singularities of algebraic varieties.
Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers. It consists mainly of algorithm design and software development for the study of properties of explicitly given algebraic varieties.

Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's Last Theorem is an example of the power of this approach.

Basic notions

Zeros of simultaneous polynomials

In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R³ could be defined as the set of all points with
A "slanted" circle in R³ can be defined as the set of all points which satisfy the two polynomial equations

Affine varieties

First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed. We consider the affine space of dimension n over k, denoted Aⁿ. When one fixes a coordinate system, one may identify Aⁿ with kⁿ. The purpose of not working with kⁿ is to emphasize that one "forgets" the vector space structure that kⁿ carries.
A function f : Aⁿ → A¹ is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in Aⁿ. The property of a function to be polynomial does not depend on the choice of a coordinate system in Aⁿ.
When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the regular functions on Aⁿ is a ring, which is denoted k.
We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k. The vanishing set of S is the set V of all points in Aⁿ where every polynomial in S vanishes. Symbolically,
A subset of Aⁿ which is V, for some S, is called an algebraic set. The V stands for variety.
Given a subset U of Aⁿ, can one recover the set of polynomials which generate it? If U is any subset of Aⁿ, define I to be the set of all polynomials whose vanishing set contains U. The I stands for ideal: if two polynomials f and g both vanish on U, then f+''g vanishes on U'', and if h is any polynomial, then hf vanishes on U, so I is always an ideal of the polynomial ring k.
Two natural questions to ask are:

Given a subset U of Aⁿ, when is U = V?
Given a set S of polynomials, when is S = I?

The answer to the first question is provided by introducing the Zariski topology, a topology on Aⁿ whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of k. Then U = V if and only if U is an algebraic set or equivalently a Zariski-closed set. The answer to the second question is given by Hilbert's Nullstellensatz. In one of its forms, it says that I is the radical of the ideal generated by S. In more abstract language, there is a Galois connection, giving rise to two closure operators; they can be identified, and naturally play a basic role in the theory; the example is elaborated at Galois connection.
For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U. Hilbert's basis theorem implies that ideals in k are always finitely generated.
An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the irreducible components of the algebraic set. An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a prime ideal of the polynomial ring.
Some authors do not make a clear distinction between algebraic sets and varieties and use irreducible variety to make the distinction when needed.

Regular functions

Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions or polynomial functions. A regular function on an algebraic set V contained in Aⁿ is the restriction to V of a regular function on Aⁿ. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even analytic.
It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space.
Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k. This ring is called the coordinate ring of V.
Since regular functions on V come from regular functions on Aⁿ, there is a relationship between the coordinate rings. Specifically, if a regular function on V is the restriction of two functions f and g in k, then f − g is a polynomial function which is null on V and thus belongs to I. Thus k may be identified with k/''I''.