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 polynomials. When there is more than one variable, geometric considerations enter and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

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 three-dimensional sphere in three-dimensional Euclidean space

\mathbb R^3

could be defined as the set of all points

(x,y,z)

with

x^2+y^2+z^2-1=0

A "slanted" circle in $\mathbb R^3$ can be defined as the set of all points $(x,y,z)$ which satisfy the two polynomial equations

x^2+y^2+z^2-1=0

x+y+z=0

Affine varieties

First we start with a field k. In classical algebraic geometry, this field was always

\mathbb C

, the complex numbers, but many of the same results are true if we assume only that k is algebraically closed. We define

{\mathbb A}^n_k

, called the affine n-space over k, to be

k^n

. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that

k^n

carries. Abstractly speaking,

{\mathbb A}^n_k

is, for the moment, just a collection of points.

Henceforth we will drop the k in ${\mathbb A}^n_k$ and instead write ${\mathbb A}^n$ .

Define a function

f:{\mathbb A}^n\to{\mathbb A}^1

to be regular if it can be written as a polynomial, that is, if there is a polynomial p in

k

such that for each point

(t_1,...,t_n)

of ${\mathbb A}^n$ ,

f(t_1,...,t_n) = p(t_1,...,t_n)

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 ${\mathbb A}^n$ as $k A}^n$ .

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 $k A}^n$ . The vanishing set of S (or vanishing locus) is the set V(S) of all points in $\mathbb{A}^n$ where every polynomial in S vanishes. In other words,

V(S) = \{(t_1,...,t_n) | \mbox{for all }p \mbox{ in }S, p(t_1,...,t_n) = 0\}

A subset of ${\mathbb A}^n$ 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 V of ${\mathbb A}^n$ which is a variety, can one recover the set of polynomials which generate it? If V is any subset of ${\mathbb A}^n$ , define I(V) to be the set of all polynomials whose vanishing set contains V. The I stands for ideal: if two polynomials f and g both vanish on V, then f+g vanishes on V, and if h is any polynomial, then hf vanishes on V, so I(V) is always an ideal of $k A}^n$ .

Two natural questions to ask are: given a subset V of ${\mathbb A}^n$ , when is

V = V(I(V))

Given a set S of polynomials, when is

S = I(V(S))

The answer to the first question is provided by introducing the Zariski topology, a topology on ${\mathbb A}^n$ which directly reflects the algebraic structure of $k A}^n$ . Then $V = V(I(V))$ , if and only if V 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.

For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set V. Hilbert's Basis Theorem implies that ideals in $k A}^n$ 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.

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. A regular function on an algebraic set V contained in

{\mathbb A}^n

is defined to be the restriction of a regular function on

{\mathbb A}^n

, 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^*. This ring is called the coordinate ring of V.

Since regular functions on V come from regular functions on ${\mathbb A}^n$ , there should be a relationship between their coordinate rings. Specifically, to get a function in kwe took a function in $k$ ^*" target="_blank" >is the quotient $k[{\mathbb A}^n/I(V)$ .

The category of affine varieties

Using regular functions from an affine variety to

{\mathbb A}^1

, 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

{\mathbb A}^n

. Choose m regular functions on V, and call them f₁, ..., f_m. We define a regular function f from V to

{\mathbb A}^m

by letting f(t₁, ..., t_n) = (f₁, ..., f_m). In other words, each f_i determines one coordinate of the range of f.

If V' is a variety contained in ${\mathbb A}^m$ , 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:

The category of affine varieties is the opposite category to the category of finitely generated reduced k-algebras and their homomorphisms.

Projective space

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.

The modern viewpoint

The modern approach to algebraic geometry redefines the basic objects. Varieties are subsumed in Alexander Grothendieck's concept of a scheme. Schemes start with the observation that if finitely generated reduced k-algebras are geometrical objects, then perhaps arbitrary commutative rings should also be geometrical objects. As such, schemes become both a more general algebro-geometric object, and a convenient language to describe those objects. This language of schemes has proved to be a valuable way of dealing with geometric concepts and has become a cornerstone of modern algebraic geometry.

Notes and history

Algebraic geometry was largely developed by Islamic mathematicians, particularly the Persian mathematician/poet Omar Khayyám (born 1048). He was well known for inventing the general method of solving cubic equations by intersecting a parabola with a circle. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of parallels which made their way to England, where they contributed to the eventual development of non-Euclidean geometry. Omar Khayyám also combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means.

Algebraic geometry was further developed by the Italian geometers in the early part of the 20th century. Enriques classified algebraic surfaces up to birational isomorphism. The style of the Italian school was very intuitive and does not meet the modern standards of rigor.

By the 1930s and 1940s, Oscar Zariski, André Weil and others realized that algebraic geometry needed to be rebuilt on foundations of commutative algebra and valuation theory. Commutative algebra (earlier known as elimination theory and then ideal theory, and refounded as the study of commutative rings and their modules) had been and was being developed by David Hilbert, Max Noether, Emanuel Lasker, Emmy Noether, Wolfgang Krull, and others. For a while there was no standard foundation for algebraic geometry.

In the 1950s and 1960s Jean-Pierre Serre and Alexander Grothendieck recast the foundations making use of the theory 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.

References

A classical textbook, predating schemes:

Modern textbooks that do not use the language of schemes:

Textbooks and references for schemes:

On the internet:

Kevin R. Coombes: Algebraic Geometry: A Total Hypertext Online System
Algebraic geometry entry on PlanetMath
Algebraic Equations and Systems of Algebraic Equations at EqWorld: The World of Mathematical Equations