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abstract variety, algebra, algebraic geometry, algebraic variety, closed, college, geometry, grad school, graduate school, math, mathematics, open, planet math, post-secondary mathematics, projective, projective space, projective variety, qualifying exam, quasi-affine variety, quasi-projective variety, set, subset, topology, university, wikipedia, wolfram, zariski topology

**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

*Each element*

**A**.^{n}**P**=(a

_{1}…,a

_{n}of

*is a called a*

**A**^{n}*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*

**(**is defined to be equal to

**P**)**(a**

**P**_{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**is defined by taking the closed sets to be the algebraic sets.

^{n}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**-{0,…,0} defined by the following equivalence relation:

^{n+1}**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**, define

^{n}*f*(

**e**) thusly: select some point

**P**in

**A**whose equivalence class is

^{n+1}**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

**are known as**

**A**^{n}*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

**with the Zariski topology, and define a**

**P**^{n}*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*.]