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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. 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.

1 Zeroes 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 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

2 Affine varieties

First we start with a field k. In classical algebraic geometry, this field was always C, the complex numbers, but many of the same results are true if we assume only that k is algebraically closed. We define , called affine n-space over k, to be kⁿ. This may seem to be useless notation, but the purpose of this definition is to forget about the vector space structure that kⁿ carries. Abstractly speaking, is, for the moment, just a collection of points.

Henceforward we will drop the k in and instead write .

Define a function

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

k[x₁,...,x_n]

such that for each point

(t₁,...,t_n)

of ,

f(t₁,...,t_n) = p(t₁,...,t_n).

Regular functions on affine space n-space are thus exactly the same as polynomials over k in n variables. We will write the regular functions on as .

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 . The vanishing set of S (or vanishing locus) is the set V(S) of all points in where every polynomial in S vanishes. In other words,

V(S)={(t₁,...,t_n) | for all p in S, p(t₁,...,t_n) = 0}.

A subset of which is V(S) for some S is called an algebraic set. The V stands for variety, which is a specific type of algebraic set defined below.

Given a subset V of which is a variety, can one recover set of polynomials which generates it? If V is any subset of , 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 .

Two natural questions to ask are: given a subset V of , 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 introducting the Zariski topology, a topology on which directly reflects the algebraic structure of . 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 connectionIn mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets ("posets"). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. here, giving rise to two closure operatorIn mathematics, given a partially ordered set P ≤), a closure operator on P is a function C : P → P with the following properties: if x ≤ y then C ''x ≤ C ''y , i. C is montonically increasing x ≤ C ''x for all x C ''C ''x ) C ''x for alls; 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 are always finitely generated.

A 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 idealIn mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. Prime ideals have a simpler description for commutative rings, so we consider this case separately below. This article on of the polynomial ring.

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