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This article is about algebraic varieties. For the term "a variety of algebras", and an explanation of the difference between a variety of algebras and an algebraic variety, see variety (universal algebra).

In mathematics, an algebraic variety is essentially a (finite or infinite) set of points where a polynomial (in one or more variables) attains, or a set of such polynomials all attain, a value of zero. Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry.

The word "variety" is employed in the sense of a mathematical manifold, for which, in Romance languages, cognates of the word "variety" are used.

Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a polynomial in one variable over the complex numbers is determined by the set of its roots, which is an inherently geometric object. Building on this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and subsets of affine space. Using the Nullstellensatz and related results, we are able to capture the geometric notion of a variety in algebraic terms as well as bring geometry to bear on questions of ring theory.

Contents 1 Formal definitions 1.1 Affine varieties 1.2 Projective varieties 2 Basic results 3 Discussion and generalizations 4 Isomorphism of algebraic varieties 5 Algebraic manifolds 6 See also 7 References

Algebraic varieties can be classed into four kinds: affine varieties, quasi-affine varieties, projective varieties, and quasi-projective varieties. There also exists the more general notion of an abstract algebraic variety.

Let k be an algebraically closed field and let Aⁿ be an affine n-space over k. The polynomials f in the ring k[x₁, ..., x_n] can be viewed as k-valued functions on Aⁿ by evaluating f at the points in Aⁿ. For each subset S of k[x₁, ..., x_n], define the zero-locus of S to be the set of points in Aⁿ on which the functions in S vanish:

A subset V of Aⁿ is called an affine algebraic set if V = Z(S) for some S. A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets. An irreducible affine algebraic set is called an affine variety.

Affine varieties can be given a natural topology, called the Zariski topology, by declaring all algebraic sets to be closed.

Given a subset V of Aⁿ, let I(V) be the ideal of all functions vanishing on V:

For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.

Let Pⁿ be a projective n-space over k. Homogeneous polynomials in k[x₀, x₁, ..., x_n] can be viewed as k-valued functions on Pⁿ by evaluating them on homogeneous coordinates. The homogeneity of the polynomial ensures that this construction is well-defined. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pⁿ on which the functions in S vanish:

A subset V of Pⁿ is called an projective algebraic set if V = Z(S) for some S. An irreducible projective algebraic set is called a projective variety.

Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.

Given a subset V of Pⁿ, let I(V) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

• An affine algebraic set V is a variety if and only if I(V) is a prime ideal; equivalently, V is a variety if and only if its coordinate ring is an integral domain.
• Every nonempty affine algebraic set may be written uniquely as a union of algebraic varieties (where none of the sets in the decomposition are subsets of each other).
• Let k[V] be the coordinate ring of the variety V. Then the dimension of V is the transcendence degree of the field of fractions of k[V] over k.

The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more — for example, to deal with varieties over fields that are not algebraically closed — some foundational changes are required. The current notion of a variety is considerably more abstract than the one above, though equivalent in the case of varieties over algebraically closed fields. An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings. A scheme is a locally ringed space such that every point has a neighbourhood, which, as a locally ringed space, is isomorphic to a spectrum of a ring. Basically, a variety is a scheme whose structure sheaf is a sheaf of K-algebras with the property that the rings R that occur above are all domains and are all finitely generated K-algebras, i.e., quotients of polynomial algebras by prime ideals.

This definition works over any field K. It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space. This also leads to problems since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled. These are usually not considered varieties, and we get rid of them by requiring the schemes underlying a variety to be separated. (There is strictly speaking also a third condition, namely, that in the definition above one needs only finitely many affine patches.)

Some modern researchers also remove the restriction on a variety having integral domain affine charts, and when speaking of a variety simply mean that the affine charts have trivial nilradical.

A complete variety is a variety such that any map from an open subset of a nonsingular curve into it can be extended uniquely to the whole curve. Every projective variety is complete, but not vice versa.

These varieties have been called 'varieties in the sense of Serre', since Serre's foundational paper FAC on sheaf cohomology was written for them. They remain typical objects to start studying in algebraic geometry, even if more general objects are also used in an auxiliary way.

One way that leads to generalisations is to allow reducible algebraic sets (and fields K that aren't algebraically closed), so the rings R may not be integral domains. This is not a big step technically. More serious is to allow nilpotents in the sheaf of rings. A nilpotent in a field must be 0: these if allowed in co-ordinate rings aren't seen as co-ordinate functions.

From the categorical point of view, nilpotents must be allowed, in order to have finite limits of varieties (to get fiber products). Geometrically this says that fibres of good mappings may have 'infinitesimal' structure. In the theory of schemes of Grothendieck these points are all reconciled: but the general scheme is far from having the immediate geometric content of a variety.

There are further generalizations called stacks and algebraic spaces.

Let V₁ and V₂ be algebraic varieties. We say that V₁ and V₂ are isomorphic, and write V₁ ≅ V₂, if there are regular maps φ : V₁ → V₂ and ψ : V₂ → V₁ such that the compositions ψ ° φ and φ ° ψ are the identity maps on V₁ and V₂ respectively.

Main article: Algebraic manifold

An algebraic manifold is an algebraic variety which is also a m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to k^m. Equivalently, the variety is smooth (free from singular points). When k is the real numbers, R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. Projective algebraic manifolds are an equivalent definition for projective varieties. The Riemann sphere is one example.

• function field
• dimension of an algebraic variety
• singular point of an algebraic variety
• birational geometry
• abelian variety
• motive
• scheme

• Robin Hartshorne (1977). Algebraic Geometry. Springer-Verlag. ISBN 0-387-90244-9. 
• David Cox; John Little, Don O'Shea (1997). Ideals, Varieties, and Algorithms, second edition, Springer-Verlag. ISBN 0-387-94680-2. 
• David Eisenbud (1999). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag. ISBN 0-387-94269-6. 
• David Dummit; Richard Foote (2003). Abstract Algebra, third edition, Wiley. ISBN 0-471-43334-9. 

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This article incorporates material from Isomorphism of varieties on PlanetMath, which is licensed under the GFDL.