Algebraic variety

Builds on Algebraically closed field Affine variety Projective variety Quasi-projective variety Irreducible component

Definition

Let $k$ be an algebraically closed field. An algebraic variety over $k$ (in the classical sense) is one of the following, with its Zariski topology:

an affine variety, that is, an irreducible algebraic set in $k^n$ ;
a quasi-affine variety, that is, a nonempty open subset of an affine variety;
a projective variety, that is, an irreducible projective algebraic set in $\mathbb{P}^n$ ;
a quasi-projective variety, that is, a nonempty open subset of a projective variety.

Remarks

Each of the four kinds is a quasi-projective variety, so the definition says that a variety is a quasi-projective variety. Every variety is irreducible and its topology is Noetherian.

Example: the affine line $k$ , the projective plane $\mathbb{P}^2$ , and the punctured plane $k^2 \setminus \set{(0, 0)}$ are varieties.

Non-example: $V(xy) \subseteq k^2$ is not a variety in this sense, since it is reducible.

Other conventions exist. Some authors allow reducible varieties. In the language of schemes, a variety over any field $k$ is often an integral separated scheme of finite type over $k$ . This includes abstract varieties that are not quasi-projective.

Reading path

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Tag	`CECPTP66EK`
URI	`m2://CECPTP66EK`
Kind	definition
Writing	W0 stub
MSC	14A10
Defines	Algebraic variety
Uses	Algebraically closed field, Affine variety, Projective variety, Quasi-projective variety, Irreducible component

Algebraic variety

Definition

Remarks

Reading path

Further reading

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