Existence of non-quasiprojective abstract algebraic varieties

One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata.^[3] Nagata's example was not complete (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 construct a toric variety that is not quasi-projective but complete.^[5]

Example 1

Let k = C, and A² be the two-dimensional affine space over C. Polynomials in the ring C[x, y] can be viewed as complex valued functions on A² by evaluating at the points in A². Let subset S of C[x, y] contain a single element  f  (x, y):

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

The zero-locus of  f  (x, y) is the set of points in A² on which this function vanishes: it is the set of all pairs of complex numbers (x, y) such that y = 1 − x. 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 ):

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

Thus the subset V = Z( f ) of A² is an 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.

Example 2

Let k = C, and A² be the two-dimensional affine space over C. Polynomials in the ring C[x, y] can be viewed as complex valued functions on A² by evaluating at the points in A². Let subset S of C[x, y] contain a single element g(x, y):

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

The zero-locus of g(x, y) is the set of points in A² on which this function vanishes, that is the set of points (x,y) such that x² + y² = 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.

Example 3

The following example is neither a hypersurface, nor a linear space, nor a single point. Let A³ be the three-dimensional affine space over C. The set of points (x, x², x³) for x in C is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane.^{[note 3]} It is the twisted cubic shown in the above figure. It may be defined by the equations

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

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) 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 n-by-n matrices over the base field k can be identified with the affine n²-space ${\displaystyle \mathbb {A} ^{n^{2}}}$ with coordinates ${\displaystyle x_{ij}}$ such that ${\displaystyle x_{ij}(A)}$ is the (i, j)-th entry of the matrix ${\displaystyle A}$. The determinant ${\displaystyle \det }$ is then a polynomial in ${\displaystyle x_{ij}}$ and thus defines the hypersurface ${\displaystyle H=V(\det )}$ in ${\displaystyle \mathbb {A} ^{n^{2}}}$. The complement of ${\displaystyle H}$ is then an open subset of ${\displaystyle \mathbb {A} ^{n^{2}}}$ that consists of all the invertible n-by-n matrices, the general linear group ${\displaystyle \operatorname {GL} _{n}(k)}$. It is an affine variety, since, in general, the complement of a hypersurface in an affine variety is affine. Explicitly, consider ${\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}}$ where the affine line is given coordinate t. Then ${\displaystyle \operatorname {GL} _{n}(k)}$ amounts to the zero-locus in ${\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}}$ of the polynomial in ${\displaystyle x_{ij},t}$:

${\displaystyle t\cdot \det[x_{ij}]-1,}$

i.e., the set of matrices A such that ${\displaystyle t\det(A)=1}$ has a solution. This is best seen algebraically: the coordinate ring of ${\displaystyle \operatorname {GL} _{n}(k)}$ is the localization ${\displaystyle k[x_{ij}\mid 0\leq i,j\leq n][{\det }^{-1}]}$, which can be identified with ${\displaystyle k[x_{ij},t\mid 0\leq i,j\leq n]/(t\det -1)}$.

The multiplicative group k^* of the base field k is the same as ${\displaystyle \operatorname {GL} _{1}(k)}$ and thus is an affine variety. A finite product of it ${\displaystyle (k^{*})^{r}}$ 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 A be a not-necessarily-commutative algebra over a field k. Even if A is not commutative, it can still happen that A has a ${\displaystyle \mathbb {Z} }$-filtration so that the associated ring ${\displaystyle \operatorname {gr} A=\bigoplus _{i=-\infty }^{\infty }A_{i}/{A_{i-1}}}$ is commutative, reduced and finitely generated as a k-algebra; i.e., ${\displaystyle \operatorname {gr} A}$ is the coordinate ring of an affine (reducible) variety X. For example, if A is the universal enveloping algebra of a finite-dimensional Lie algebra ${\displaystyle {\mathfrak {g}}}$, then ${\displaystyle \operatorname {gr} A}$ is a polynomial ring (the PBW theorem); more precisely, the coordinate ring of the dual vector space ${\displaystyle {\mathfrak {g}}^{*}}$.

Let M be a filtered module over A (i.e., ${\displaystyle A_{i}M_{j}\subset M_{i+j}}$). If ${\displaystyle \operatorname {gr} M}$ is fintiely generated as a ${\displaystyle \operatorname {gr} A}$-algebra, then the support of ${\displaystyle \operatorname {gr} M}$ in X; i.e., the locus where ${\displaystyle \operatorname {gr} M}$ does not vanish is called the characteristic variety of M.^[6] The notion plays an important role in the theory of D-modules.

Example 1

The affine plane curve y² = 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. The projective line P¹ is an example of a projective curve; it can be viewed as the curve in the projective plane P² = {[x, y, z]} defined by x = 0. For another example, first consider the affine cubic curve

${\displaystyle y^{2}=x^{3}-x.}$

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

${\displaystyle y^{2}z=x^{3}-xz^{2},}$

which defines a curve in P² called an elliptic curve. The curve has genus one (genus formula); in particular, it is not isomorphic to the projective line P¹, 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 V be a finite-dimensional vector space. The Grassmannian variety G_n(V) is the set of all n-dimensional subspaces of V. It is a projective variety: it is embedded into a projective space via the Plücker embedding:

${\displaystyle {\begin{cases}G_{n}(V)\hookrightarrow \mathbf {P} \left(\wedge ^{n}V\right)\\\langle b_{1},\ldots ,b_{n}\rangle \mapsto [b_{1}\wedge \cdots \wedge b_{n}]\end{cases}}}$

where b_i are any set of linearly independent vectors in V, ${\displaystyle \wedge ^{n}V}$ is the n-th exterior power of V, and the bracket [w] means the line spanned by the nonzero vector w.

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 C be a smooth complete curve and ${\displaystyle \operatorname {Pic} (C)}$ the Picard group of it; i.e., the group of isomorphism classes of line bundles on C. Since C is smooth, ${\displaystyle \operatorname {Pic} (C)}$ can be identified as the divisor class group of C and thus there is the degree homomorphism ${\displaystyle \operatorname {deg}$ :\operatorname {Pic} (C)\to \mathbb {Z} } . The Jacobian variety ${\displaystyle \operatorname {Jac} (C)}$ of C is the kernel of this degree map; i.e., the group of the divisor classes on C 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, ${\displaystyle \operatorname {Jac} (C)}$ is a projective variety. The tangent space to ${\displaystyle \operatorname {Jac} (C)}$ at the identity element is naturally isomorphic to ${\displaystyle \operatorname {H} ^{1}(C,{\mathcal {O}}_{C});}$^[7] hence, the dimension of ${\displaystyle \operatorname {Jac} (C)}$ is the genus of ${\displaystyle C}$.

Fix a point ${\displaystyle P_{0}}$ on ${\displaystyle C}$. For each integer ${\displaystyle n>0}$, there is a natural morphism^[8]

${\displaystyle C^{n}\to \operatorname {Jac} (C),\,(P_{1},\dots ,P_{r})\mapsto [P_{1}+\cdots +P_{n}-nP_{0}]}$

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

Moduli varieties

Given an integer ${\displaystyle g\geq 0}$, the set of isomorphism classes of smooth complete curves of genus ${\displaystyle g}$ is called the moduli of curves of genus ${\displaystyle g}$ and is denoted as ${\displaystyle {\mathfrak {M}}_{g}}$. There are few ways to show this moduli has a structure of a possibly reducible algebraic variety; for example, one way is to use geometric 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 a stable curve of genus ${\displaystyle g\geq 2}$, a not-necessarily-smooth complete curve with no terribly bad singularities and not-so-large automorphism group. The moduli of stable curves ${\displaystyle {\overline {\mathfrak {M}}}_{g}}$, the set of isomorphism classes of stable curves of genus ${\displaystyle g\geq 2}$, is then a projective variety which contains ${\displaystyle {\mathfrak {M}}_{g}}$ as an open subset. Since ${\displaystyle {\overline {\mathfrak {M}}}_{g}}$ is obtained by adding boundary points to ${\displaystyle {\mathfrak {M}}_{g}}$, ${\displaystyle {\overline {\mathfrak {M}}}_{g}}$ is colloquially said to be a compactification of ${\displaystyle {\mathfrak {M}}_{g}}$. Historically a paper of Mumford and Deligne^[10] introduced the notion of a stable curve to show ${\displaystyle {\mathfrak {M}}_{g}}$ is irreducible when ${\displaystyle g\geq 2}$.

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 of stable and semistable vector bundles on a smooth complete curve ${\displaystyle C}$. The moduli of semistable vector bundles of a given rank ${\displaystyle n}$ and a given degree ${\displaystyle d}$ (degree of the determinant of the bundle) is then a projective variety denoted as ${\displaystyle SU_{C}(n,d)}$, which contains the set ${\displaystyle U_{C}(n,d)}$ of isomorphism classes of stable vector bundles of rank ${\displaystyle n}$ and degree ${\displaystyle d}$ as an open subset.^[11] Since a line bundle is stable, such a moduli is a generalization of the Jacobian variety of ${\displaystyle C}$.

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 over ${\displaystyle \mathbb {C} }$ is the problem of compactifying ${\displaystyle D/\Gamma }$, the quotient of a bounded symmetric domain ${\displaystyle D}$ by an action of an arithmetic discrete group ${\displaystyle \Gamma }$.^[12] A basic example of ${\displaystyle D/\Gamma }$ is when ${\displaystyle D={\mathfrak {H}}_{g}}$, Siegel's upper half-space and ${\displaystyle \Gamma }$ commensurable with ${\displaystyle \operatorname {Sp} (2g,\mathbb {Z} )}$; in that case, ${\displaystyle D/\Gamma }$ has an interpretation as the moduli ${\displaystyle {\mathfrak {A}}_{g}}$ of principally polarized complex abelian varieties of dimension ${\displaystyle g}$ (a principal polarization identifies an abelian variety with its dual). The theory of toric varieties (or torus embeddings) gives a way to compactify ${\displaystyle D/\Gamma }$, a toroidal compactification of it.^[13][14] But there are other ways to compactify ${\displaystyle D/\Gamma }$; for example, there is the minimal compactification of ${\displaystyle D/\Gamma }$ 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;^[15] 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.