The fundamental objects of study in algebraic geometry are algebraic varieties, geometric manifestations of solutions of systems of polynomial equations. Plane algebraic curves, which include lines, circles, parabolas, lemniscates, and Cassini ovals, form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and relations between the curves given by different equations.
Descartes's idea of coordinates is central to algebraic geometry, but it has undergone a series of remarkable transformations beginning in the early 19th century. Before then, the coordinates were assumed to be tuples of real numbers, but first complex numbers, and then elements of an arbitrary field became acceptable. Homogeneous coordinates of projective geometry offered an extension of the notion of coordinate system in a different direction, and enriched the scope of algebraic geometry. Much of the development 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 and complex geometry. One key distinction between classical projective geometry of 19th century and modern algebraic geometry, in the form given to it by Grothendieck and Serre, is that the former is concerned with the more geometric notion of a point, while the latter emphasizes the more analytic concepts of a regular function and a regular map and extensively draws on sheaf theory. Another important difference lies in the scope of the subject. Grothendieck's idea of scheme provides the language and the tools for geometric treatment of arbitrary commutative rings and, in particular, bridges algebraic geometry with algebraic number theory. Andrew Wiles's celebrated proof of Fermat's last theorem is a vivid testament to the power of this approach. André Weil, Grothendieck, and Deligne also demonstrated that the fundamental ideas of topology of manifolds have deep analogues in algebraic geometry over finite fields.
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 in three-dimensional Euclidean space R3 could be defined as the set of all points (x,y,z) with
A "slanted" circle in R3 can be defined as the set of all points (x,y,z) which satisfy the two polynomial equations
A function f : An → A1 is said to be regular if it can be written as a polynomial, that is, if there is a polynomial p in k[x1,...,xn] such that f(t1,...,tn) = p(t1,...,tn) for every point (t1,...,tn) of An.
Regular functions on affine n-space are thus exactly the same as polynomials over k in n variables. We will write the regular functions on An as k[An].
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[An]. The vanishing set of S (or vanishing locus) is the set V(S) of all points in An where every polynomial in S vanishes. In other words,
A subset of An which is V(S), for some S, is called an algebraic set. The V stands for variety (a specific type of algebraic set to be defined below).
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(U) 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(U) is always an ideal of k[An].
Two natural questions to ask are:
The answer to the first question is provided by introducing the Zariski topology, a topology on An which directly reflects the algebraic structure of k[An]. Then U = V(I(U)) if and only if U is 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(V(S)) is the prime 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[An] are always finitely generated.
An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if the polynomials defining it generate a prime ideal of the polynomial ring.
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. A regular function on an algebraic set V contained in An is defined to be the restriction of a regular function on An, in the sense we defined above.
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[V]. This ring is called the coordinate ring of V.
Since regular functions on V come from regular functions on An, there should be a relationship between their coordinate rings. Specifically, to get a function in k[V] we took a function in k[An], and we said that it was the same as another function if they gave the same values when evaluated on V. This is the same as saying that their difference is zero on V. From this we can see that k[V] is the quotient k[An]/I(V).
Using regular functions from an affine variety to A1, we can define regular functions from one affine variety to another. First we will define a regular function from a variety into affine space: Let V be a variety contained in An. Choose m regular functions on V, and call them f1, ..., fm. We define a regular function f from V to Am by letting f(t1, ..., tn) = (f1, ..., fm). In other words, each fi determines one coordinate of the range of f.
If V' is a variety contained in Am, we say that f is a regular function from V to V' if the range of f is contained in V'.
This makes the collection of all affine varieties into a category, where the objects are affine varieties and the morphisms are regular maps. The following theorem characterizes the category of affine varieties:
Consider the variety V(y - x2). If we draw it, we get a parabola. As x increases, the slope of the line from the origin to the point (x, x2) becomes larger and larger. As x decreases, the slope of the same line becomes smaller and smaller.
Compare this to the variety V(y - x3). This is a cubic equation. As x increases, the slope of the line from the origin to the point (x, x3) becomes larger and larger just as before. But unlike before, as x decreases, the slope of the same line again becomes larger and larger. So the behavior "at infinity" of V(y-x3) is different from the behavior "at infinity" of V(y - x2). It is, however, difficult to make the concept of "at infinity" meaningful, if we restrict to working in affine space.
The remedy to this is to work in projective space. Projective space has properties analogous to those of a compact Hausdorff space. Among other things, it lets us make precise the notion of "at infinity" by including extra points. The behavior of a variety at those extra points then gives us more information about it. As it turns out, V(y - x3) has a singularity at one of those extra points, but V(y - x2) is smooth.
While projective geometry was originally established on a synthetic foundation, the use of homogeneous coordinates allowed the introduction of algebraic techniques. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For this reason, projective space plays a fundamental role in algebraic geometry.
A further generalization is possible to Universal algebraic geometry in which every variety of algebra has its own algebraic geometry. The term variety of algebra should not be confused with algebraic variety.
Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of Renaissance mathematicians such as Cardano and Niccolo Fontana "Tartaglia" on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably Blaise Pascal who argued against the use of algebraic and analytical methods in geometry. The French mathematicians Franciscus Vieta and later René Descartes and Pierre de Fermat revolutionized the conventional way of thinking about construction problems through the introduction of coordinate geometry. They were interested primarily in the properties of algebraic curves, such as those defined by Diophantine equations (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes).
During the same period, Blaise Pascal and Desargues approached geometry from a different perspective, developing the synthetic notions of Projections. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek ruler and compass construction. Ultimately, the analytic geometry of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of Newton and Leibniz. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the calculus of infinitesimals of Lagrange and Euler.
In the 1950s and 1960s Jean-Pierre Serre and Alexander Grothendieck recast the foundations making use of sheaf theory. 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 new applications were made, both to number theory and to more classical geometric questions on algebraic varieties, singularities and moduli.
An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group. The prototypical examples are the elliptic curves, which have a rich theory. They 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 effective computation with concretely-given polynomials have also been developed. The most important is the technique of Gröbner bases which is employed in all computer algebra systems.
Modern textbooks that do not use the language of schemes:
Textbooks and references for schemes:
On the Internet:
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