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| {{About|algebraic varieties|the term "variety of algebras", and an explanation of the difference between a variety of algebras and an algebraic variety|variety (universal algebra)}}
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| [[Image:Twisted cubic curve.png|200px|thumb|The [[twisted cubic]] is a projective algebraic variety.]]
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| In [[mathematics]], '''algebraic varieties''' (also called '''varieties''') are one of the central objects of study in [[algebraic geometry]]. Classically, an algebraic variety was defined to be the [[solution set|set of solutions]] of a [[system of polynomial equations]], over the [[real number|real]] or [[complex number]]s. Modern definitions of an algebraic variety generalize this notion in several different ways, while attempting to preserve the geometric intuition behind the original definition.{{r|Hartshorne|page1=58}}
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| Conventions regarding the definition of an algebraic variety differ slightly. For example, some authors require that an "''algebraic variety''" is, by definition, ''irreducible'' (which means that it is not the union of two smaller sets that are closed in the [[Zariski topology]]), while others do not. When the former convention is used, non-irreducible algebraic varieties are called '''algebraic sets'''.
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| The notion of variety is similar to that of [[manifold]], the difference being that a variety may have [[singular point of an algebraic variety|singular points]], while a manifold may not. In many languages, both varieties and manifolds are named by the same word.<!--
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| Sorry for ambiguous word "many";
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| The languages that distinguish variety and manifold include: German, English, Iranian, Finn, Swedish and Chinese.
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| The languages that use the same word include: Bulgarian, Catalan, Czech, Spanish, French, Korean, Italian, Hebrew, Dutch, Japanese, Polish, Portuguese, Russian, Slovenian, Ukranian, and Vietnamese.
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| (This list is original research.)
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| I'm not sure if it's worth recording for English wikipedia. -->
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| Proven around the year 1800, the [[fundamental theorem of algebra]] establishes a link between [[algebra]] and [[geometry]] by showing that a [[monic polynomial]] in one variable with [[complex numbers|complex]] coefficients (an algebraic object) is determined by the set of its [[root of a function|root]]s (a geometric object). Generalizing this result, [[Hilbert's Nullstellensatz]] provides a fundamental correspondence between [[ideal (ring theory)|ideals]] of [[polynomial ring]]s and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of [[ring theory]]. This correspondence is the specifity of [[algebraic geometry]] among the other subareas of [[geometry]].
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| ==Introduction and definitions==<!-- This section is linked from [[Zariski topology]] -->
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| An ''affine variety'' over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way. The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s.
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| ===Affine varieties===
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| {{main|Affine variety}}
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| Let ''k'' be an [[algebraically closed field]] and let '''A'''<sup>''n''</sup> be an [[Affine space|affine ''n''-space]] over ''k''. The polynomials ƒ in the ring ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] can be viewed as ''k''-valued functions on '''A'''<sup>''n''</sup> by evaluating ƒ at the points in '''A'''<sup>''n''</sup>, i.e. by choosing values in ''A'' for each ''x<sub>i</sub>''. For each set ''S'' of polynomials in ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>], define the zero-locus ''Z''(''S'') to be the set of points in '''A'''<sup>''n''</sup> on which the functions in ''S'' simultaneously vanish, that is to say
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| :<math>Z(S) = \{x \in \mathbb A^n \mid f(x) = 0 \text{ for all } f\in S\}.</math>
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| A subset ''V'' of '''A'''<sup>''n''</sup> is called an '''affine algebraic set''' if ''V'' = ''Z''(''S'') for some ''S''.{{r|Hartshorne|page1=2}} A nonempty affine algebraic set ''V'' is called '''irreducible''' if it cannot be written as the union of two [[subset|proper]] algebraic subsets.{{r|Hartshorne|page1=3}} An irreducible affine algebraic set is also called an '''affine variety'''.{{r|Hartshorne|page1=3}} (Many authors use the phrase ''affine variety'' to refer to any affine algebraic set, irreducible or not<ref group="note">Hartshorne, p.xv, notes that his choice is not conventional; see for example, Harris, p.3</ref>)
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| Affine varieties can be given a [[natural topology]] by declaring the [[closed set]]s to be precisely the affine algebraic sets. This topology is called the [[Zariski topology]].{{r|Hartshorne|page1=2}}
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| Given a subset ''V'' of '''A'''<sup>''n''</sup>, we define ''I''(''V'') to be the ideal of all functions vanishing on ''V'':
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| :<math>I(V) = \{f \in k[x_1,\ldots,x_n] \mid f(x) = 0 \text{ for all } x\in V\}.</math>
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| For any affine algebraic set ''V'', the '''coordinate ring''' or '''structure ring''' of ''V'' is the [[quotient ring|quotient]] of the polynomial ring by this ideal.{{r|Hartshorne|page1=4}}
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| ==={{anchor|Projective variety}}Projective varieties and quasi-projective varieties===
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| {{main|Projective variety|Quasi-projective variety}}
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| Let ''k'' be an [[algebraically closed field]] and let '''P'''<sup>''n''</sup> be a [[Algebraic geometry of projective spaces|projective ''n''-space]] over ''k''. Let ''f'' ∈ ''k'' [''x''<sub>0</sub>, ..., ''x''<sub>''n''</sub>] be a [[homogeneous polynomial]] of degree ''d''. It is not well-defined to evaluate ''f'' on points in '''P'''<sup>''n''</sup> in [[homogeneous coordinates]]. However, because ''f'' is homogeneous, ''f''(''λx''<sub>0</sub>, ..., ''λx''<sub>''n''</sub>) = ''λ''<sup>''d''</sup>''f''(''x''<sub>0</sub>, ..., ''x''<sub>''n''</sub>), it ''does'' make sense to ask whether ''f'' vanishes at a point [''x''<sub>0</sub> : ... : ''x''<sub>''n''</sub>]. For each set ''S'' of homogeneous polynomials, define the zero-locus of ''S'' to be the set of points in '''P'''<sup>''n''</sup> on which the functions in ''S'' vanish:
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| :<math>Z(S) = \{x \in \mathbf{P}^n \mid f(x) = 0 \text{ for all } f\in S\}.</math>
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| A subset ''V'' of '''P'''<sup>''n''</sup> is called a '''projective algebraic set''' if ''V'' = ''Z''(''S'') for some ''S''.{{r|Hartshorne|page1=9}} An irreducible projective algebraic set is called a '''projective variety'''.{{r|Hartshorne|page1=10}}
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| Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.
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| Given a subset ''V'' of '''P'''<sup>''n''</sup>, let ''I''(''V'') be the [[ideal (ring theory)|ideal]] generated by all homogeneous polynomials vanishing on ''V''. For any projective algebraic set ''V'', the '''[[homogeneous coordinate ring|coordinate ring]]''' of ''V'' is the quotient of the polynomial ring by this ideal.{{r|Hartshorne|page1=10}}
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| A '''[[quasi-projective variety]]''' is a [[Zariski topology|Zariski open]] subset of a projective variety. Notice that every affine variety is quasi-projective.<ref>Hartshorne, Exercise I.2.9, p.12</ref> Notice also that the complement of an algebraic set in an affine variety is a quasi-projective variety; in the context of affine varieties, such a quasi-projective variety is usually not called a variety but a [[constructible set (topology)|constructible set]].
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| ==={{anchor|Abstract varieties}}Abstract varieties===
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| In classical algebraic geometry, all varieties were by definition [[quasiprojective variety|quasiprojective varieties]], meaning that they were open subvarieties of closed subvarieties of [[projective space]]. For example, in Chapter 1 of Hartshorne a ''variety'' over an algebraically closed field is defined to be a [[quasi-projective variety]],{{r|Hartshorne|page1=15}} but from Chapter 2 onwards, the term '''variety''' (also called an '''abstract variety''') refers to a more general object, which locally is a quasi-projective variety, but when viewed as a whole is not necessarily quasi-projective; i.e. it might not have an embedding into [[projective space]].{{r|Hartshorne|page1=105}} So classically the definition of an algebraic variety required an embedding into projective space, and this embedding was used to define the topology on the variety and the [[regular function]]s on the variety. The disadvantage of such a definition is that not all varieties come with natural embeddings into projective space. For example, under this definition, the product '''P'''<sup>1</sup>×'''P'''<sup>1</sup> is not a variety until it is embedded into the projective space; this is usually done by the [[Segre embedding]]. However, any variety which admits one embedding into projective space admits many others by composing the embedding with the [[Veronese embedding]]. Consequently many notions which should be intrinsic, such as the concept of a regular function, are not obviously so.
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| The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made by [[André Weil]]. In his ''Foundations of Algebraic Geometry'', Weil defined an abstract algebraic variety using [[valuation (algebra)|valuation]]s. [[Claude Chevalley]] made a definition of a scheme which served a similar purpose, but was more general. However, it was [[Alexander Grothendieck]]'s definition of a [[scheme (mathematics)|scheme]] that was both most general and found the most widespread acceptance. In Grothendieck's language, an abstract algebraic variety is usually defined to be an [[Glossary of scheme theory#integral|integral]], [[Glossary of scheme theory#Separated and proper morphisms|separated]] [[Scheme (mathematics)|scheme]] of [[Glossary of scheme theory#Finite, quasi-finite, and finite type morphisms|finite type]] over an [[algebraically closed field]],<ref group=note>{{Harvnb|Hartshorne|1976|pp=104–105}}</ref> although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed.<ref group=note>Liu, Qing. ''Algebraic Geometry and Arithmetic Curves'', p. 55 Definition 2.3.47, and p. 88 Example 3.2.3</ref> Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field.
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| ====Existence of non-quasiprojective abstract algebraic varieties====
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| One of the earliest examples of a non-quasiprojective algebraic variety were given by [[Masayoshi Nagata|Nagata]].<ref name=Nagata56/> Nagata's example was not [[complete variety|complete]] (the analog of compactness), but soon afterwards he found an [[algebraic surface]] which was complete and non-projective.<ref name=Nagata57/> Since then other examples have been found.
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| ==Examples==
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| ===Subvariety===
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| A '''subvariety''' is a subset of a variety which is itself a variety.
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| ===Affine algebraic variety===
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| ====Example 1====
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| Let ''k'' be the field of complex numbers '''C'''. Let '''A'''<sup>2</sup> be a two dimensional [[affine space]] over '''C'''. A polynomial ƒ in the ring '''C'''[''x'', ''y''] can be viewed as complex valued functions on '''A'''<sup>2</sup> by evaluating ƒ at the points in '''A'''<sup>2</sup>. Let subset ''S'' of '''C'''[''x'', ''y''] contain a single element ƒ(''x'', ''y''):
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| :<math>f(x, y) = x+y-1. \, </math>
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| The zero-locus of ƒ(''x'', ''y'') is the set of points in '''A'''<sup>2</sup> on which this function vanishes: it is the set of all pairs of complex numbers (''x'',''y'') such that ''y'' = 1 − ''x'', commonly known as a [[line (geometry)|line]]. This is the set ''Z''(ƒ):
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| :<math>Z(f) = \{ (x,1-x) \in \mathbf{C}^2 \}.</math>
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| Thus the subset ''V'' = ''Z''(ƒ) of '''A'''<sup>2</sup> is an [[Algebraic variety#Affine varieties|algebraic set]]. The set ''V'' is not empty. It is irreducible, as it cannot be written as the union of two proper algebraic subsets. Thus it is an affine algebraic variety.
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| ====Example 2====
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| Let again ''k'' be the field of complex numbers '''C'''. Let '''A'''<sup>2</sup> be a two dimensional affine space over '''C'''. The polynomials ''g'' in the ring '''C'''[''x'', ''y''] can be viewed as complex valued functions on '''A'''<sup>2</sup> by evaluating ''g'' at the points in '''A'''<sup>2</sup>. Let subset ''S'' of '''C'''[''x'', ''y''] contain a single element ''g''(''x'', ''y''):
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| :<math>g(x, y) = x^2 + y^2 - 1. \, </math>
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| The zero-locus of ''g''(''x'', ''y'') is the set of points in '''A'''<sup>2</sup> on which this function vanishes, that is the set of points (''x'',''y'') such that ''x<sup>2</sup>'' + ''y<sup>2</sup>'' = 1. As ''g''(''x'', ''y'') is an [[absolutely irreducible]] polynomial, this is an algebraic variety. The set of its real points (that is the points for which ''x'' and ''y'' are real numbers), is known as the [[unit circle]]; this name is also often given to the whole variety.
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| ====Example 3====
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| The following example is neither a [[hypersurface]], nor a [[vector space|linear space]], nor a single point. Let '''A'''<sup>3</sup> be the three dimensional affine space over '''C'''. The set of points (''x'', ''x''<sup>2</sup>, ''x''<sup>3</sup>) for ''x'' in '''C''' is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane.<ref group="note">Harris, p.9; that it is irreducible is stated as an exercise in Hartshorne p.7</ref> It is the [[twisted cubic]] shown in the above figure. It may be defined by the equations
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| : <math>
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| \begin{align}
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| y-x^2&=0\\
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| z-x^3&=0. \,
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| \end{align} </math>
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| The fact that the set of the solutions of this system of equations is irreducible needs a proof. The simplest results from the fact that the projection (''x'', ''y'', ''z'') → (''x'', ''y'') is [[injective function|injective]] on the set of the solutions and that its image is an irreducible plane curve.
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| For more difficult examples, a similar proof may always be given, but may imply a difficult computation: first a [[Gröbner basis]] computation to compute the [[dimension of an algebraic variety|dimension]], followed by a random linear change of variables (not always needed); then a [[Gröbner basis]] computation for another [[monomial order]]ing to compute the projection and to prove that it is injective, and finally a [[polynomial factorization]] to prove the irreducibility of the image.
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| ==Basic results==
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| * 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]].{{r|Harris|page1=52}}{{r|Hartshorne|page1=4}}
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| * Every nonempty affine algebraic set may be written uniquely as a finite union of algebraic varieties (where none of the varieties in the decomposition is a subvariety of any other).{{r|Hartshorne|page1=5}}
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| * 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''.{{r|Harris|page1=135}}
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| == Isomorphism of algebraic varieties ==
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| Let V<sub>1</sub> and V<sub>2</sub> be algebraic varieties. We say that V<sub>1</sub> and V<sub>2</sub> are [[graph isomorphism|isomorphic]], and write V<sub>1</sub> ≅ V<sub>2</sub>, if there are [[regular function|regular map]]s φ : V<sub>1</sub> → V<sub>2</sub> and ψ : V<sub>2</sub> → V<sub>1</sub> such that the [[function (mathematics)|compositions]] ψ ° φ and φ ° ψ are the [[identity function|identity maps]] on V<sub>1</sub> and V<sub>2</sub> respectively.
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| ==Discussion and generalizations==
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| {{no footnotes|date=March 2013|section}}
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| 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 field|algebraically closed]] — some foundational changes are required. The modern 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 (mathematics)|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 over ''k'' is a scheme whose [[structure sheaf]] is a [[sheaf (mathematics)|sheaf]] of ''k''-algebras with the property that the rings ''R'' that occur above are all [[integral domain]]s and are all finitely generated ''k''-algebras, that is to say, they are quotients of [[polynomial algebra]]s by [[prime ideal]]s.
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| This definition works over any field ''k''. It allows you to glue affine varieties (along common open sets) without
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| worrying whether the resulting object can be put into some projective space. This also leads to difficulties since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled. Such objects are usually not considered varieties, and are eliminated by requiring the schemes underlying a variety to be ''separated''. (Strictly speaking, there is also a third condition, namely, that one needs only finitely many affine patches in the definition above.)
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| Some modern researchers also remove the restriction on a variety having [[integral domain]] affine charts, and when speaking of a variety only require that the affine charts have trivial [[nilradical of a ring|nilradical]].
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| A [[complete variety]] is a variety such that any map from an open subset of a nonsingular [[algebraic curve|curve]] into it can be extended uniquely to the whole curve. Every projective variety is complete, but not ''vice versa''.
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| These varieties have been called 'varieties in the sense of Serre', since [[Jean-Pierre Serre|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.
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| 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. A more significant modification is to allow [[nilpotent]]s in the sheaf of rings. A nilpotent in a field must be 0: these if allowed in coordinate rings aren't seen as ''coordinate functions''.
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| From the [[category theory|categorical]] point of view, nilpotents must be allowed, in order to have finite limits of varieties (to get [[fiber product]]s).{{citation needed|date=August 2012}} Geometrically this says that fibres of good mappings may have 'infinitesimal' structure. In the theory of [[scheme (mathematics)|scheme]]s of [[Alexander Grothendieck|Grothendieck]] these points are all reconciled: but the general ''scheme'' is far from having the immediate geometric content of a ''variety''.
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| There are further generalizations called [[algebraic space]]s and [[algebraic stack|stack]]s.
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| ==Algebraic manifolds==
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| {{main|Algebraic manifold}}
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| An algebraic manifold is an algebraic variety which is also an ''m''-dimensional [[manifold]], and hence every sufficiently small local patch is isomorphic to ''k''<sup>''m''</sup>. Equivalently, the variety is [[smooth function|smooth]] (free from [[singular point of an algebraic variety|singular points]]). When ''k'' is the [[real number]]s, '''R''', algebraic manifolds are called [[Nash manifold]]s. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. [[Projective algebraic manifold]]s are an equivalent definition for projective varieties. The [[Riemann sphere]] is one example.
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| ==See also==
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| *[[Variety (disambiguation)]]
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| *[[Function field of an algebraic variety]]
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| *[[Dimension of an algebraic variety]]
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| *[[Singular point of an algebraic variety]]
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| *[[Birational geometry]]
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| *[[Abelian variety]]
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| *[[Motive (algebraic geometry)|Motive]]
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| *[[Scheme (mathematics)|Scheme]]
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| *[[Analytic variety]]
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| *[[Zariski–Riemann space]]
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| *[[Semi-algebraic set]]
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| ==Footnotes==
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| {{reflist|group=note}}
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| ==References==
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| {{reflist|refs=
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| <ref name="Harris">
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| {{cite book| last=Harris|first=Joe|authorlink = Joe Harris (mathematician)|year=1992|title=Algebraic Geometry - A first course|publisher= [[Springer Science+Business Media|Springer-Verlag]]| isbn=0-387-97716-3}}
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| </ref>
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| <ref name=Nagata56>
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| {{Citation | last1=Nagata | first1=Masayoshi | author1-link=Masayoshi Nagata | title=On the imbedding problem of abstract varieties in projective varieties | mr=0088035 | year=1956 | journal=Memoirs of the College of Science, University of Kyoto. Series A: Mathematics | volume=30 | pages=71–82}}
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| </ref> | |
| <ref name=Nagata57> | |
| {{Citation | last1=Nagata | first1=Masayoshi | author1-link=Masayoshi Nagata | title=On the imbeddings of abstract surfaces in projective varieties | mr=0094358 | year=1957 | journal=Memoirs of the College of Science, University of Kyoto. Series A: Mathematics | volume=30 | pages=231–235}}
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| </ref>
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| <ref name="Hartshorne">
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| {{cite book
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| | last = Hartshorne
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| | first = Robin
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| | authorlink = Robin Hartshorne
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| | year = 1977
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| | title = Algebraic Geometry
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| | publisher = [[Springer Science+Business Media|Springer-Verlag]]
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| | isbn = 0-387-90244-9
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| }}
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| </ref>
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| }}
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| {{reflist}}
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| *{{cite book
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| | last = Cox
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| | first = David
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| | authorlink = David Cox (mathematician)
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| | coauthors = John Little, Don O'Shea
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| | year = 1997
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| | title = Ideals, Varieties, and Algorithms
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| | edition = second
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| | publisher = [[Springer Science+Business Media|Springer-Verlag]]
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| | isbn = 0-387-94680-2
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| }}
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| *{{cite book
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| | last = Eisenbud
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| | first = David
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| | authorlink = David Eisenbud
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| | year = 1999
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| | title = Commutative Algebra with a View Toward Algebraic Geometry
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| | publisher = [[Springer Science+Business Media|Springer-Verlag]]
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| | isbn = 0-387-94269-6
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| }}
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| * {{Cite web
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| | last=Milne
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| | first=James S.
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| |authorlink=James Milne (mathematician)
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| | title=Algebraic Geometry
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| | year=2008
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| | url=http://www.jmilne.org/math/CourseNotes/ag.html
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| | accessdate=2009-09-01
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| | postscript=.
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| }}
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| {{PlanetMath attribution|id=6838|title=Isomorphism of varieties}}
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| [[Category:Algebraic geometry]]
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| [[Category:Algebraic varieties|*]]
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