**Preliminary Definitions**

Let **k** be an algebraically closed field.* Affine n-space *over **k** is the set of all sets (a_{1},…,a_{n}) of n elements of **k***, *and is denoted by** A**^{n}_{k} or just** A**^{n}. Each element **P**=(a_{1}…,a_{n} of **A**^{n} is a called a *point*, and the a_{i} are called its *coordinates*.

Suppose *f* is a member of the polynomial ring A in n variables over **k** and **P**=(a_{1},…,a_{n}) is a point in **A**^{n}. Then *f***(****P**) is defined to be equal to** ****P**(a_{1},…,a_{n}).

Let T be some subset of A. The set Z(T) of zeros of T is defined to be the greatest subset of elements **P** of **A**^{n} such that *f*(**P**)=0 for all *f* in T. The sets Z(T) are called *algebraic sets*, and the *Zariski topology* on **A**^{n} is defined by taking the closed sets to be the algebraic sets.

A subset of a topological space is *irreducible* if it is nonempty and contains no two proper closed subsets of which it can be written as the union.

*Projective n-space* **P**^{n} over an algebraically closed field is defined as the set of equivalence classes of elements of **A**^{n+1}-{0,…,0} defined by the following equivalence relation: **P**_{1} is in the same equivalence class as **P**_{2} iff **P**_{1}=*L***P**_{2} for some *L* in **k**-{0}. For any member of the polynomial ring in n+1 variables over **k** and any member **e** of **P**^{n}, define *f*(**e**) thusly: select some point **P** in **A**^{n+1} whose equivalence class is **e**. Then compute *f*(**e**). This value will be determined up to multiplication by a nonzero element of **k**, which means that nonzero values of *f*(**e**) are undefined. Convention remedies this unfortunate situation by defining the nonzero values to be equal to 1.

**Varieties**

*Varieties* fall into four types, as detailed below.

Let **A**^{n} have the Zariski topology. The closed irreducible subsets of** ****A**^{n} are known as *affine algebraic varieties* or just *algebraic varieties*, and *quasi-affine algebraic variety* or *quasi-affine variety* is an open subset of an affine variety.

Let T be some subset of the polynomial ring in n+1 variables over **k**. Z(T) is then defined to be be the set of all equivalence classes in **P**^{n} such that *f*(**e**)**=**0 for all *f* in T, and the Zariski topology can be defined in the same way as for affine space. Let us endow** ****P**^{n} with the Zariski topology, and define a *projective algebraic variety* or *projective variety* (respectively, *quasi-projective algebraic variety* or *quasi-projective variety*) to be an irreducible closed (respectively, an open subset of an irreducible closed) subset of** ****P**^{n}.

[Note: there also exists the (obviously) less concrete notion of an *abstract variety*.]