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Algebraic variety

From Wikipedia, the free encyclopedia
(Redirected fromAlgebraic set)
Mathematical object studied in the field of algebraic geometry
Not to be confused withVariety of algebras.
Thetwisted cubic is a projective algebraic variety. It is the intersection of the surfacesy=x2{\displaystyle y=x^{2}} andz=x3{\displaystyle z=x^{3}}.

Algebraic varieties are the central objects of study inalgebraic geometry, a sub-field ofmathematics. Classically, an algebraic variety is defined as theset of solutions of asystem of polynomial equations over thereal orcomplex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.[1]: 58 

Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to beirreducible, which means that it is not theunion of two smallersets that areclosed in theZariski topology. Under this definition, non-irreducible algebraic varieties are calledalgebraic sets. Other conventions do not require irreducibility.

Thefundamental theorem of algebra establishes a link betweenalgebra andgeometry by showing that amonic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of itsroots (a geometric object) in thecomplex plane. Generalizing this result,Hilbert's Nullstellensatz provides a fundamental correspondence betweenideals ofpolynomial rings and algebraic sets. Using theNullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions ofring theory. This correspondence is a defining feature of algebraic geometry.

Many algebraic varieties aredifferentiable manifolds, but an algebraic variety may havesingular points while a differentiable manifold cannot. Algebraic varieties can be characterized by theirdimension. Algebraic varieties of dimension one are calledalgebraic curves and algebraic varieties of dimension two are calledalgebraic surfaces.

In the context of modernscheme theory, an algebraic variety over afield is an integral (irreducible and reduced) scheme over that field whosestructure morphism is separated and of finite type.

Overview and definitions

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Anaffine variety over analgebraically 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, butNagata gave an example of such a new variety in the 1950s.

Affine varieties

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Main article:Affine variety

For an algebraically closed fieldK and anatural numbern, letAn be anaffinen-space overK, identified toKn{\displaystyle K^{n}} through the choice of anaffine coordinate system. The polynomialsf in the ringK[x1, ...,xn] can be viewed asK-valued functions onAn by evaluatingf at the points inAn, i.e. by choosing values inK for eachxi. For each setS of polynomials inK[x1, ...,xn], define the zero-locusZ(S) to be the set of points inAn on which the functions inS simultaneously vanish, that is to say

Z(S)={x∈An∣f(x)=0 for all f∈S}.{\displaystyle Z(S)=\left\{x\in \mathbf {A} ^{n}\mid f(x)=0{\text{ for all }}f\in S\right\}.}

A subsetV ofAn is called anaffine algebraic set ifV =Z(S) for someS.[1]: 2  A nonempty affine algebraic setV is calledirreducible if it cannot be written as the union of twoproper algebraic subsets.[1]: 3  An irreducible affine algebraic set is also called anaffine variety.[1]: 3  (Some authors use the phraseaffine variety to refer to any affine algebraic set, irreducible or not.[note 1])

Affine varieties can be given anatural topology by declaring theclosed sets to be precisely the affine algebraic sets. This topology is called the Zariski topology.[1]: 2 

Given a subsetV ofAn, we defineI(V) to be the ideal of all polynomial functions vanishing onV:

I(V)={f∈K[x1,…,xn]∣f(x)=0 for all x∈V}.{\displaystyle I(V)=\left\{f\in K[x_{1},\ldots ,x_{n}]\mid f(x)=0{\text{ for all }}x\in V\right\}.}

For any affine algebraic setV, thecoordinate ring orstructure ring ofV is thequotient of the polynomial ring by this ideal.[1]: 4 

Projective varieties and quasi-projective varieties

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Main articles:Projective variety andQuasi-projective variety

Letk be an algebraically closed field and letPn be theprojectiven-space overk. Letf ink[x0, ...,xn] be ahomogeneous polynomial of degreed. It is not well-defined to evaluatef on points inPn inhomogeneous coordinates. However, becausef is homogeneous, meaning thatf(λx0, ...,λxn) =λdf(x0, ...,xn), itdoes make sense to ask whetherf vanishes at a point[x0 : ... :xn]. For each setS of homogeneous polynomials, define the zero-locus ofS to be the set of points inPn on which the functions inS vanish:

Z(S)={x∈Pn∣f(x)=0 for all f∈S}.{\displaystyle Z(S)=\{x\in \mathbf {P} ^{n}\mid f(x)=0{\text{ for all }}f\in S\}.}

A subsetV ofPn is called aprojective algebraic set ifV =Z(S) for someS.[1]: 9  An irreducible projective algebraic set is called aprojective variety.[1]: 10 

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

Given a subsetV ofPn, letI(V) be the ideal generated by all homogeneous polynomials vanishing onV. For any projective algebraic setV, thecoordinate ring ofV is the quotient of the polynomial ring by this ideal.[1]: 10 

Aquasi-projective variety is aZariski open subset of a projective variety. Notice that every affine variety is quasi-projective usingx0=0{\displaystyle x_{0}=0} chart.[2] 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 aconstructible set.

Abstract varieties

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In classical algebraic geometry, all varieties were by definitionquasi-projective varieties, meaning that they were open subvarieties of closed subvarieties of aprojective space. For example, in Chapter 1 ofHartshorne avariety over an algebraically closed field is defined to be aquasi-projective variety,[1]: 15  but from Chapter 2 onwards, the termvariety (also called anabstract 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 intoprojective space.[1]: 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 theregular functions 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 productP1 ×P1 is not a variety until it is embedded into a larger projective space; this is usually done by theSegre embedding. Furthermore, any variety that admits one embedding into projective space admits many others, for example by composing the embedding with theVeronese embedding; thus many notions that should be intrinsic, such as that of a regular function, are not obviously so.

The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made byAndré Weil in hisFoundations of Algebraic Geometry, usingvaluations.Claude Chevalley made a definition of ascheme, which served a similar purpose, but was more general. However,Alexander Grothendieck's definition of a scheme is more general still and has received the most widespread acceptance. In Grothendieck's language, an abstract algebraic variety is usually defined to be anintegral,separated scheme offinite type over an algebraically closed field,[1]: 104–105  although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed.[note 2] Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field.

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 Nagata.[3] Nagata's example was notcomplete (the analog of compactness), but soon afterwards he found an algebraic surface that was complete and non-projective.[4][1]: Remark 4.10.2 p.105  Since then other examples have been found: for example, it is straightforward to constructtoric varieties that are not quasi-projective but complete.[5]

Examples

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Subvariety

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Asubvariety is a subset of a variety that is itself a variety (with respect to the topological structure induced by the ambient variety). For example, every open subset of a variety is a variety. See alsoclosed immersion.

Hilbert's Nullstellensatz says that closed subvarieties of an affine or projective variety are in one-to-one correspondence with the prime ideals or non-irrelevant homogeneous prime ideals of the coordinate ring of the variety.

Affine variety

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Example 1

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Letk =C, andA2 be the two-dimensionalaffine space overC. Polynomials in the ringC[x,y] can be viewed as complex valued functions onA2 by evaluating at the points inA2. Let subsetS ofC[x,y] contain a single elementf(x,y):

f(x,y)=x+y−1.{\displaystyle f(x,y)=x+y-1.}

The zero-locus off(x,y) is the set of points inA2 on which this function vanishes: it is the set of all pairs of complex numbers (x,y) such thaty = 1 −x. This is called aline in the affine plane. (In theclassical topology coming from the topology on the complex numbers, a complex line is a real manifold of dimension two.) This is the setZ(f):

Z(f)={(x,1−x)∈C2}.{\displaystyle Z(f)=\{(x,1-x)\in \mathbf {C} ^{2}\}.}

Thus the subsetV =Z(f) ofA2 is analgebraic set. The setV 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.

Example 2

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Letk =C, andA2 be the two-dimensional affine space overC. Polynomials in the ringC[x,y] can be viewed as complex valued functions onA2 by evaluating at the points inA2. Let subsetS ofC[x,y] contain a single elementg(x,y):

g(x,y)=x2+y2−1.{\displaystyle g(x,y)=x^{2}+y^{2}-1.}

The zero-locus ofg(x,y) is the set of points inA2 on which this function vanishes, that is the set of points (x,y) such thatx2 +y2 = 1. Asg(x,y) is anabsolutely irreducible polynomial, this is an algebraic variety. The set of its real points (that is the points for whichx andy are real numbers), is known as theunit circle; this name is also often given to the whole variety.

Example 3

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The following example is neither ahypersurface, nor alinear space, nor a single point. LetA3 be the three-dimensional affine space overC. The set of points (x,x2,x3) forx inC is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane.[note 3] It is thetwisted cubic shown in the above figure. It may be defined by the equations

y−x2=0z−x3=0{\displaystyle {\begin{aligned}y-x^{2}&=0\\z-x^{3}&=0\end{aligned}}}

The irreducibility of this algebraic set needs a proof. One approach in this case is to check that the projection (x,y,z) → (x,y) isinjective on the set of the solutions and that its image is an irreducible plane curve.

For more difficult examples, a similar proof may always be given, but may imply a difficult computation: first aGröbner basis computation to compute the dimension, followed by a random linear change of variables (not always needed); then aGröbner basis computation for anothermonomial ordering to compute the projection and to prove that it isgenerically injective and that its image is ahypersurface, and finally apolynomial factorization to prove the irreducibility of the image.

General linear group

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The set ofn-by-n matrices over the base fieldk can be identified with the affinen2-spaceAn2{\displaystyle \mathbb {A} ^{n^{2}}} with coordinatesxij{\displaystyle x_{ij}} such thatxij(A){\displaystyle x_{ij}(A)} is the (i,j)-th entry of the matrixA{\displaystyle A}. Thedeterminantdet{\displaystyle \det } is then a polynomial inxij{\displaystyle x_{ij}} and thus defines the hypersurfaceH=V(det){\displaystyle H=V(\det )} inAn2{\displaystyle \mathbb {A} ^{n^{2}}}. The complement ofH{\displaystyle H} is then an open subset ofAn2{\displaystyle \mathbb {A} ^{n^{2}}} that consists of all the invertiblen-by-n matrices, thegeneral linear groupGLn⁡(k){\displaystyle \operatorname {GL} _{n}(k)}. It is an affine variety, since, in general, the complement of a hypersurface in an affine variety is affine. Explicitly, considerAn2×A1{\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}} where the affine line is given coordinatet. ThenGLn⁡(k){\displaystyle \operatorname {GL} _{n}(k)} amounts to the zero-locus inAn2×A1{\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}} of the polynomial inxij,t{\displaystyle x_{ij},t}:

t⋅det[xij]−1,{\displaystyle t\cdot \det[x_{ij}]-1,}

i.e., the set of matricesA such thattdet(A)=1{\displaystyle t\det(A)=1} has a solution. This is best seen algebraically: the coordinate ring ofGLn⁡(k){\displaystyle \operatorname {GL} _{n}(k)} is thelocalizationk[xij∣0≤i,j≤n][det−1]{\displaystyle k[x_{ij}\mid 0\leq i,j\leq n][{\det }^{-1}]}, which can be identified withk[xij,t∣0≤i,j≤n]/(tdet−1){\displaystyle k[x_{ij},t\mid 0\leq i,j\leq n]/(t\det -1)}.

The multiplicative group k* of the base fieldk is the same asGL1⁡(k){\displaystyle \operatorname {GL} _{1}(k)} and thus is an affine variety. A finite product of it(k∗)r{\displaystyle (k^{*})^{r}} is analgebraic torus, which is again an affine variety.

A general linear group is an example of alinear algebraic group, an affine variety that has a structure of agroup in such a way the group operations are morphism of varieties.

Characteristic variety

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Main article:Characteristic variety

LetA be a not-necessarily-commutative algebra over a fieldk. Even ifA is not commutative, it can still happen thatA has aZ{\displaystyle \mathbb {Z} }-filtration so that theassociated ringgr⁡A=⨁i=−∞∞Ai/Ai−1{\displaystyle \operatorname {gr} A=\bigoplus _{i=-\infty }^{\infty }A_{i}/{A_{i-1}}} is commutative, reduced and finitely generated as ak-algebra; i.e.,gr⁡A{\displaystyle \operatorname {gr} A} is the coordinate ring of an affine (reducible) varietyX. For example, ifA is theuniversal enveloping algebra of a finite-dimensionalLie algebrag{\displaystyle {\mathfrak {g}}}, thengr⁡A{\displaystyle \operatorname {gr} A} is a polynomial ring (thePBW theorem); more precisely, the coordinate ring of the dual vector spaceg∗{\displaystyle {\mathfrak {g}}^{*}}.

LetM be a filtered module overA (i.e.,AiMj⊂Mi+j{\displaystyle A_{i}M_{j}\subset M_{i+j}}). Ifgr⁡M{\displaystyle \operatorname {gr} M} is fintiely generated as agr⁡A{\displaystyle \operatorname {gr} A}-algebra, then thesupport ofgr⁡M{\displaystyle \operatorname {gr} M} inX; i.e., the locus wheregr⁡M{\displaystyle \operatorname {gr} M} does not vanish is called thecharacteristic variety ofM.[6] The notion plays an important role in the theory ofD-modules.

Projective variety

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Aprojective variety is a closed subvariety of a projective space. That is, it is the zero locus of a set ofhomogeneous polynomials that generate aprime ideal.

Example 1

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The affine plane curvey2 =x3 −x. The corresponding projective curve is called an elliptic curve.

A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. Theprojective lineP1 is an example of a projective curve; it can be viewed as the curve in the projective planeP2 = {[x,y,z]} defined byx = 0. For another example, first consider the affine cubic curve

y2=x3−x.{\displaystyle y^{2}=x^{3}-x.}

in the 2-dimensional affine space (over a field of characteristic not two). It has the associated cubic homogeneous polynomial equation:

y2z=x3−xz2,{\displaystyle y^{2}z=x^{3}-xz^{2},}

which defines a curve inP2 called anelliptic curve. The curve has genus one (genus formula); in particular, it is not isomorphic to the projective lineP1, which has genus zero. Using genus to distinguish curves is very basic: in fact, the genus is the first invariant one uses to classify curves (see also the construction ofmoduli of algebraic curves).

Example 2: Grassmannian

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LetV be a finite-dimensional vector space. TheGrassmannian varietyGn(V) is the set of alln-dimensional subspaces ofV. It is a projective variety: it is embedded into a projective space via thePlücker embedding:

{Gn(V)↪P(∧nV)⟨b1,…,bn⟩↦[b1∧⋯∧bn]{\displaystyle {\begin{cases}G_{n}(V)\hookrightarrow \mathbf {P} \left(\wedge ^{n}V\right)\\\langle b_{1},\ldots ,b_{n}\rangle \mapsto [b_{1}\wedge \cdots \wedge b_{n}]\end{cases}}}

wherebi are any set of linearly independent vectors inV,∧nV{\displaystyle \wedge ^{n}V} is then-thexterior power ofV, and the bracket [w] means the line spanned by the nonzero vectorw.

The Grassmannian variety comes with a naturalvector bundle (orlocally free sheaf in other terminology) called thetautological bundle, which is important in the study ofcharacteristic classes such asChern classes.

Jacobian variety and abelian variety

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LetC be a smooth complete curve andPic⁡(C){\displaystyle \operatorname {Pic} (C)} thePicard group of it; i.e., the group of isomorphism classes of line bundles onC. SinceC is smooth,Pic⁡(C){\displaystyle \operatorname {Pic} (C)} can be identified as thedivisor class group ofC and thus there is the degree homomorphismdeg:Pic⁡(C)→Z{\displaystyle \operatorname {deg} :\operatorname {Pic} (C)\to \mathbb {Z} }. TheJacobian varietyJac⁡(C){\displaystyle \operatorname {Jac} (C)} ofC is the kernel of this degree map; i.e., the group of the divisor classes onC of degree zero. A Jacobian variety is an example of anabelian variety, a complete variety with a compatibleabelian group structure on it (the name "abelian" is however not because it is an abelian group). An abelian variety turns out to be projective (in short, algebraictheta functions give an embedding into a projective space. Seeequations defining abelian varieties); thus,Jac⁡(C){\displaystyle \operatorname {Jac} (C)} is a projective variety. The tangent space toJac⁡(C){\displaystyle \operatorname {Jac} (C)} at the identity element is naturally isomorphic toH1⁡(C,OC);{\displaystyle \operatorname {H} ^{1}(C,{\mathcal {O}}_{C});}[7] hence, the dimension ofJac⁡(C){\displaystyle \operatorname {Jac} (C)} is the genus ofC{\displaystyle C}.

Fix a pointP0{\displaystyle P_{0}} onC{\displaystyle C}. For each integern>0{\displaystyle n>0}, there is a natural morphism[8]

Cn→Jac⁡(C),(P1,…,Pr)↦[P1+⋯+Pn−nP0]{\displaystyle C^{n}\to \operatorname {Jac} (C),\,(P_{1},\dots ,P_{r})\mapsto [P_{1}+\cdots +P_{n}-nP_{0}]}

whereCn{\displaystyle C^{n}} is the product ofn copies ofC. Forg=1{\displaystyle g=1} (i.e.,C is an elliptic curve), the above morphism forn=1{\displaystyle n=1} turns out to be an isomorphism;[1]: Ch. IV, Example 1.3.7.  in particular, an elliptic curve is an abelian variety.

Moduli varieties

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Given an integerg≥0{\displaystyle g\geq 0}, the set of isomorphism classes of smooth complete curves of genusg{\displaystyle g} is called themoduli of curves of genusg{\displaystyle g} and is denoted asMg{\displaystyle {\mathfrak {M}}_{g}}. There are few ways to show this moduli has a structure of a possibly reducible algebraic variety; for example, one way is to usegeometric invariant theory which ensures a set of isomorphism classes has a (reducible) quasi-projective variety structure.[9] Moduli such as the moduli of curves of fixed genus is typically not a projective variety; roughly the reason is that a degeneration (limit) of a smooth curve tends to be non-smooth or reducible. This leads to the notion of astable curve of genusg≥2{\displaystyle g\geq 2}, a not-necessarily-smooth complete curve with no terribly bad singularities and not-so-large automorphism group. The moduli of stable curvesM¯g{\displaystyle {\overline {\mathfrak {M}}}_{g}}, the set of isomorphism classes of stable curves of genusg≥2{\displaystyle g\geq 2}, is then a projective variety which containsMg{\displaystyle {\mathfrak {M}}_{g}} as an open dense subset. SinceM¯g{\displaystyle {\overline {\mathfrak {M}}}_{g}} is obtained by adding boundary points toMg{\displaystyle {\mathfrak {M}}_{g}},M¯g{\displaystyle {\overline {\mathfrak {M}}}_{g}} is colloquially said to be acompactification ofMg{\displaystyle {\mathfrak {M}}_{g}}. Historically a paper of Mumford and Deligne[10] introduced the notion of a stable curve to showMg{\displaystyle {\mathfrak {M}}_{g}} is irreducible wheng≥2{\displaystyle g\geq 2}.

The moduli of curves exemplifies a typical situation: a moduli of nice objects tend not to be projective but only quasi-projective. Another case is a moduli of vector bundles on a curve. Here, there are the notions ofstable and semistable vector bundles on a smooth complete curveC{\displaystyle C}. The moduli of semistable vector bundles of a given rankn{\displaystyle n} and a given degreed{\displaystyle d} (degree of the determinant of the bundle) is then a projective variety denoted asSUC(n,d){\displaystyle SU_{C}(n,d)}, which contains the setUC(n,d){\displaystyle U_{C}(n,d)} of isomorphism classes of stable vector bundles of rankn{\displaystyle n} and degreed{\displaystyle d} as an open subset.[11] Since a line bundle is stable, such a moduli is a generalization of the Jacobian variety ofC{\displaystyle C}.

In general, in contrast to the case of moduli of curves, a compactification of a moduli need not be unique and, in some cases, different non-equivalent compactifications are constructed using different methods and by different authors. An example overC{\displaystyle \mathbb {C} } is the problem of compactifyingD/Γ{\displaystyle D/\Gamma }, the quotient of a bounded symmetric domainD{\displaystyle D} by an action of an arithmetic discrete groupΓ{\displaystyle \Gamma }.[12] A basic example ofD/Γ{\displaystyle D/\Gamma } is whenD=Hg{\displaystyle D={\mathfrak {H}}_{g}},Siegel's upper half-space andΓ{\displaystyle \Gamma }commensurable withSp⁡(2g,Z){\displaystyle \operatorname {Sp} (2g,\mathbb {Z} )}; in that case,D/Γ{\displaystyle D/\Gamma } has an interpretation as the moduliAg{\displaystyle {\mathfrak {A}}_{g}} of principally polarized complex abelian varieties of dimensiong{\displaystyle g} (a principal polarization identifies an abelian variety with its dual). The theory of toric varieties (or torus embeddings) gives a way to compactifyD/Γ{\displaystyle D/\Gamma }, atoroidal compactification of it.[13][14] But there are other ways to compactifyD/Γ{\displaystyle D/\Gamma }; for example, there is theminimal compactification ofD/Γ{\displaystyle D/\Gamma } due to Baily and Borel: it is theprojective variety associated to the graded ring formed bymodular forms (in the Siegel case,Siegel modular forms;[15] see alsoSiegel modular variety). The non-uniqueness of compactifications is due to the lack of moduli interpretations of those compactifications; i.e., they do not represent (in the category-theory sense) any natural moduli problem or, in the precise language, there is no naturalmoduli stack that would be an analog of moduli stack of stable curves.

Non-affine and non-projective example

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An algebraic variety can be neither affine nor projective. To give an example, letX =P1 ×A1 andp:X →A1 the projection. HereX is an algebraic variety since it is a product of varieties. It is not affine sinceP1 is a closed subvariety ofX (as the zero locus ofp), but an affine variety cannot contain a projective variety of positive dimension as a closed subvariety. It is not projective either, since there is a nonconstantregular function onX; namely,p.

Another example of a non-affine non-projective variety isX =A2 − (0, 0) (cf.Morphism of varieties § Examples.)

Non-examples

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Consider the affine lineA1{\displaystyle \mathbb {A} ^{1}} overC{\displaystyle \mathbb {C} }. The complement of the circle{z∈C with |z|2=1}{\displaystyle \{z\in \mathbb {C} {\text{ with }}|z|^{2}=1\}} inA1=C{\displaystyle \mathbb {A} ^{1}=\mathbb {C} } is not an algebraic variety (nor even an algebraic set). Note that|z|2−1{\displaystyle |z|^{2}-1} is not a polynomial inz{\displaystyle z} (although it is a polynomial in the real coordinatesx,y{\displaystyle x,y}). On the other hand, the complement of the origin inA1=C{\displaystyle \mathbb {A} ^{1}=\mathbb {C} } is an algebraic (affine) variety, since the origin is the zero-locus ofz{\displaystyle z}. This may be explained as follows: the affine line has dimension one and so any subvariety of it other than itself must have strictly less dimension; namely, zero.

For similar reasons, aunitary group (over the complex numbers) is not an algebraic variety, while the special linear groupSLn⁡(C){\displaystyle \operatorname {SL} _{n}(\mathbb {C} )} is a closed subvariety ofGLn⁡(C){\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}, the zero-locus ofdet−1{\displaystyle \det -1}. (Over a different base field, a unitary group can however be given a structure of a variety.)

Basic results

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  • An affine algebraic setV is a varietyif and only ifI(V) is aprime ideal; equivalently,V is a variety if and only if its coordinate ring is anintegral domain.[16]: 52 [1]: 4 
  • 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).[1]: 5 
  • Thedimension of a variety may be defined in various equivalent ways. SeeDimension of an algebraic variety for details.
  • A product of finitely many algebraic varieties (over an algebraically closed field) is an algebraic variety. A finite product of affine varieties is affine[17] and a finite product of projective varieties is projective.

Isomorphism of algebraic varieties

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See also:Morphism of varieties

LetV1,V2 be algebraic varieties. We sayV1 andV2 areisomorphic, and writeV1 ≅V2, if there areregular mapsφ :V1 →V2 andψ :V2 →V1 such that thecompositionsψ ∘φ andφ ∘ψ are theidentity maps onV1 andV2 respectively.

Discussion and generalizations

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This section includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this section byintroducing more precise citations.(March 2013) (Learn how and when to remove this message)

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 notalgebraically 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. Anabstract 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 alocally ringed space such that every point has a neighbourhood that, as a locally ringed space, is isomorphic to aspectrum of a ring. Basically, a variety overk is a scheme whosestructure sheaf is asheaf ofk-algebras with the property that the ringsR that occur above are allintegral domains and are all finitely generatedk-algebras, that is to say, they are quotients ofpolynomial algebras byprime ideals.

This definition works over any fieldk. 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 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 beseparated. (Strictly speaking, there is also a third condition, namely, that one needs only finitely many affine patches in the definition above.)

Some modern researchers also remove the restriction on a variety havingintegral domain affine charts, and when speaking of a variety only require that the affine charts have trivialnilradical.

Acomplete variety is a variety such that any map from an open subset of a nonsingularcurve 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", sinceSerre's foundational paperFAC[18]onsheaf 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 generalizations is to allow reducible algebraic sets (and fieldsk that aren't algebraically closed), so the ringsR may not be integral domains. A more significant modification is to allownilpotents in the sheaf of rings, that is, rings which are notreduced. This is one of several generalizations of classical algebraic geometry that are built intoGrothendieck's theory of schemes.

Allowing nilpotent elements in rings is related to keeping track of "multiplicities" in algebraic geometry. For example, the closed subscheme of the affine line defined byx2 = 0 is different from the subscheme defined byx = 0 (the origin). More generally, thefiber of a morphism of schemesX →Y at a point ofY may be non-reduced, even ifX andY are reduced. Geometrically, this says that fibers of good mappings may have nontrivial "infinitesimal" structure.

There are further generalizations calledalgebraic spaces andstacks.

Algebraic manifolds

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Main article:Algebraic manifold

An algebraic manifold is an algebraic variety that is also anm-dimensional manifold, and hence every sufficiently small local patch is isomorphic tokm. Equivalently, the variety issmooth (free from singular points). Whenk is the real numbers,R, algebraic manifolds are calledNash 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. TheRiemann sphere is one example.

See also

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Notes

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  1. ^Hartshorne, p.xv, Harris, p.3
  2. ^Liu, Qing.Algebraic Geometry and Arithmetic Curves, p. 55 Definition 2.3.47, and p. 88 Example 3.2.3
  3. ^Harris, p.9; that it is irreducible is stated as an exercise in Hartshorne p.7

References

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  1. ^abcdefghijklmnopHartshorne, Robin (1977).Algebraic Geometry.Springer-Verlag.ISBN 0-387-90244-9.
  2. ^Hartshorne, Exercise I.2.9, p.12
  3. ^Nagata, Masayoshi (1956)."On the imbedding problem of abstract varieties in projective varieties".Memoirs of the College of Science, University of Kyoto. Series A: Mathematics.30:71–82.doi:10.1215/kjm/1250777138.MR 0088035.
  4. ^Nagata, Masayoshi (1957)."On the imbeddings of abstract surfaces in projective varieties".Memoirs of the College of Science, University of Kyoto. Series A: Mathematics.30 (3):231–235.doi:10.1215/kjm/1250777007.MR 0094358.S2CID 118328992.
  5. ^In page 65 ofFulton, William (1993),Introduction to toric varieties,Princeton University Press,ISBN 978-0-691-00049-7, a remark describes a complete toric variety that has no non-trivial line bundle; thus, in particular, it has no ample line bundle.
  6. ^Definition 1.1.12 in Ginzburg, V., 1998. Lectures on D-modules. University of Chicago.
  7. ^Milne 2008, Proposition 2.1.
  8. ^Milne 2008, The beginning of § 5.
  9. ^MFK 1994, Theorem 5.11.
  10. ^Deligne, Pierre;Mumford, David (1969)."The irreducibility of the space of curves of given genus"(PDF).Publications Mathématiques de l'IHÉS.36:75–109.CiteSeerX 10.1.1.589.288.doi:10.1007/bf02684599.S2CID 16482150.
  11. ^MFK 1994, Appendix C to Ch. 5.
  12. ^Mark Goresky. Compactifications and cohomology of modular varieties. In Harmonic analysis,the trace formula, and Shimura varieties, volume 4 of Clay Math. Proc., pages 551–582. Amer.Math. Soc., Providence, RI, 2005.
  13. ^Ash, A.;Mumford, David; Rapoport, M.; Tai, Y. (1975),Smooth compactification of locally symmetric varieties(PDF), Brookline, Mass.: Math. Sci. Press,ISBN 978-0-521-73955-9,MR 0457437
  14. ^Namikawa, Yukihiko (1980).Toroidal Compactification of Siegel Spaces. Lecture Notes in Mathematics. Vol. 812.doi:10.1007/BFb0091051.ISBN 978-3-540-10021-8.
  15. ^Chai, Ching-Li (1986). "Siegel Moduli Schemes and Their Compactifications overC{\displaystyle \mathbb {C} }".Arithmetic Geometry. pp. 231–251.doi:10.1007/978-1-4613-8655-1_9.ISBN 978-1-4613-8657-5.
  16. ^Harris, Joe (1992).Algebraic Geometry - A first course. Graduate Texts in Mathematics. Vol. 133.Springer-Verlag.doi:10.1007/978-1-4757-2189-8.ISBN 0-387-97716-3.
  17. ^Algebraic Geometry I. Encyclopaedia of Mathematical Sciences. Vol. 23. 1994.doi:10.1007/978-3-642-57878-6.ISBN 978-3-540-63705-9.
  18. ^Serre, Jean-Pierre (1955)."Faisceaux Algebriques Coherents"(PDF).Annals of Mathematics.61 (2):197–278.doi:10.2307/1969915.JSTOR 1969915.

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