Subvariety A
subvariety 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 also
closed 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 Example 1 Let , and be the two-dimensional
affine space over . Polynomials in the ring can be viewed as complex valued functions on by evaluating at the points in . Let subset of contain a single element : : f(x, y) = x+y-1. The zero-locus of is the set of points in on which this function vanishes: it is the set of all pairs of complex numbers such that . This is called a
line in the affine plane. (In the
classical topology coming from the topology on the complex numbers, a complex line is a real manifold of dimension two.) This is the set : : Z(f) = \{ (x,1-x) \in \mathbf{C}^2 \}. Thus the subset of is an
algebraic set. The set 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 Let , and be the two-dimensional affine space over . Polynomials in the ring can be viewed as complex valued functions on by evaluating at the points in . Let subset of contain a single element : : g(x, y) = x^2 + y^2 - 1. The zero-locus of is the set of points in on which this function vanishes, that is the set of points such that . As is an
absolutely irreducible polynomial, this is an algebraic variety. The set of its real points (that is the points for which and are real numbers), is known as the
unit circle; this name is also often given to the whole variety.
Example 3 The following example is neither a
hypersurface, nor a
linear space, nor a single point. Let be the three-dimensional affine space over . The set of points for in is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane. It is the
twisted cubic shown in the above figure. It may be defined by the equations : \begin{align} y-x^2&=0\\ z-x^3&=0 \end{align} The irreducibility of this algebraic set needs a proof. One approach in this case is to check that the projection is
injective 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 a
Gröbner basis computation to compute the dimension, followed by a random linear change of variables (not always needed); then a
Gröbner basis computation for another
monomial ordering to compute the projection and to prove that it is
generically injective and that its image is a
hypersurface, and finally a
polynomial factorization to prove the irreducibility of the image.
General linear group The set of -by- matrices over the base field can be identified with the affine -space with coordinates such that is the th entry of the matrix . The
determinant is then a polynomial in and thus defines the hypersurface in . The complement of is then an open subset of that consists of all the invertible -by- matrices, the
general linear group . It is an affine variety, since, in general, the complement of a hypersurface in an affine variety is affine. Explicitly, consider , where the affine line is given coordinate . Then amounts to the zero-locus in of the polynomial in , : : t \cdot \det[x_{ij}] - 1, i.e., the set of matrices such that has a solution. This is best seen algebraically: the coordinate ring of is the
localization , which can be identified with . The multiplicative group of the base field is the same as and thus is an affine variety. A finite product of it is an
algebraic torus, which is again an affine variety. A general linear group is an example of a
linear algebraic group, an affine variety that has a structure of a
group in such a way the group operations are morphism of varieties.
Characteristic variety Let be a not-necessarily-commutative algebra over a field . Even if is not commutative, it can still happen that has a -filtration so that the
associated ring \textstyle \operatorname{gr} A = \bigoplus_{i=-\infty}^{\infty} A_i/{A_{i-1}} is commutative, reduced and finitely generated as a -algebra; i.e., is the coordinate ring of an affine (reducible) variety . For example, if is the
universal enveloping algebra of a finite-dimensional
Lie algebra \mathfrak g, then is a polynomial ring (the
PBW theorem); more precisely, the coordinate ring of the dual vector space \mathfrak g^*. Let be a filtered module over (i.e., ). If is finitely generated as a -algebra, then the
support of in ; i.e., the locus where does not vanish is called the
characteristic variety of . The notion plays an important role in the theory of
-modules.
Projective variety A
projective variety is a closed subvariety of a projective space. That is, it is the zero locus of a set of
homogeneous polynomials that generate a
prime ideal.
Example 1 A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. The
projective line is an example of a projective curve; it can be viewed as the curve in the projective plane defined by . For another example, first consider the affine cubic curve : 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: : y^2z = x^3 - xz^2, which defines a curve in called an
elliptic curve. The curve has genus one (
genus formula); in particular, it is not isomorphic to the projective line , 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 of
moduli of algebraic curves).
Example 2: Grassmannian Let be a finite-dimensional vector space. The
Grassmannian variety is the set of all -dimensional subspaces of . It is a projective variety: it is embedded into a projective space via the
Plücker embedding: : \begin{cases} G_n(V) \hookrightarrow \mathbf{P} \left (\bigwedge^n V \right ) \\ \langle b_1, \ldots, b_n \rangle \mapsto [b_1 \wedge \cdots \wedge b_n] \end{cases} where are any set of linearly independent vectors in , \wedge^n V is the th
exterior power of , and the bracket means the line spanned by the nonzero vector . The Grassmannian variety comes with a natural
vector bundle (or
locally free sheaf in other terminology) called the
tautological bundle, which is important in the study of
characteristic classes such as
Chern classes.
Jacobian variety and abelian variety Let be a smooth complete curve and the
Picard group of it; i.e., the group of isomorphism classes of line bundles on . Since is smooth, can be identified as the
divisor class group of and thus there is the degree homomorphism . The
Jacobian variety of is the kernel of this degree map; i.e., the group of the divisor classes on of degree zero. A Jacobian variety is an example of an
abelian variety, a complete variety with a compatible
abelian 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, algebraic
theta functions give an embedding into a projective space. See
equations defining abelian varieties); thus, is a projective variety. The tangent space to at the identity element is naturally isomorphic to \operatorname{H}^1(C, \mathcal{O}_C); hence, the dimension of is the genus of . Fix a point on . For each integer , there is a natural morphism : C^n \to \operatorname{Jac}(C), \, (P_1, \dots, P_r) \mapsto [P_1 + \cdots + P_n - nP_0] where is the product of copies of . For (i.e., is an elliptic curve), the above morphism for turns out to be an isomorphism; in particular, an elliptic curve is an abelian variety.
Moduli varieties Given an integer , the set of isomorphism classes of smooth complete curves of genus is called the
moduli of curves of genus g and is denoted as \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 use
geometric invariant theory which ensures a set of isomorphism classes has a (reducible) quasi-projective variety structure. 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 a
stable curve of genus , a not-necessarily-smooth complete curve with no terribly bad singularities and not-so-large automorphism group. The moduli of stable curves \overline{\mathfrak{M}}_g, the set of isomorphism classes of stable curves of genus , is then a projective variety which contains \mathfrak{M}_g as an open dense subset. Since \overline{\mathfrak{M}}_g is obtained by adding boundary points to \mathfrak{M}_g, \overline{\mathfrak{M}}_g is colloquially said to be a
compactification of \mathfrak{M}_g. Historically a paper of Mumford and Deligne introduced the notion of a stable curve to show \mathfrak{M}_g is irreducible when . 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 of
stable and semistable vector bundles on a smooth complete curve . The moduli of semistable vector bundles of a given rank and a given degree (degree of the determinant of the bundle) is then a projective variety denoted as SU_C(n, d), which contains the set U_C(n, d) of isomorphism classes of stable vector bundles of rank and degree as an open subset. Since a line bundle is stable, such a moduli is a generalization of the Jacobian variety of . 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 over is the problem of compactifying , the quotient of a bounded symmetric domain by an action of an arithmetic discrete group . A basic example of D / \Gamma is when D = \mathfrak{H}_g,
Siegel's upper half-space and
commensurable with ; in that case, has an interpretation as the moduli \mathfrak{A}_g of principally polarized complex abelian varieties of dimension (a principal polarization identifies an abelian variety with its dual). The theory of toric varieties (or torus embeddings) gives a way to compactify , a
toroidal compactification of it. But there are other ways to compactify ; for example, there is the
minimal compactification of due to Baily and Borel: it is the
projective variety associated to the graded ring formed by
modular forms (in the Siegel case,
Siegel modular forms; see also
Siegel 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 natural
moduli stack that would be an analog of moduli stack of stable curves.
Non-affine and non-projective example An algebraic variety can be neither affine nor projective. To give an example, let and the projection. Here
X is an algebraic variety since it is a product of varieties. It is not affine since is a closed subvariety of (as the zero locus of ), 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 nonconstant
regular function on ; namely, . Another example of a non-affine non-projective variety is (cf. ''''.)
Non-examples Consider the affine line over . The complement of the circle in is not an algebraic variety (nor even an algebraic set). Note that is not a polynomial in (although it is a polynomial in the real coordinates , ). On the other hand, the complement of the origin in is an algebraic (affine) variety, since the origin is the zero-locus of . 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, a
unitary group (over the complex numbers) is not an algebraic variety, while the special linear group is a closed subvariety of , the zero-locus of . (Over a different base field, a unitary group can however be given a structure of a variety.) == Basic results ==