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OR:

Algebraic varieties are the central objects of study in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a sub-field of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. Classically, an algebraic variety is defined as the set of solutions of a
system of polynomial equations A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some field . A ''solution'' of a polynomial system is a set of values for the ...
over the real or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynom ...
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
by showing that a
monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\c ...
(an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by t ...
. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variab ...
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 In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...
. This correspondence is a defining feature of algebraic geometry. Many algebraic varieties are
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s, but an algebraic variety may have singular points while a manifold cannot. Algebraic varieties can be characterized by their
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
. Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces. In the context of modern scheme theory, an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field whose
structure morphism In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A c ...
is separated and of finite type.

# Overview and definitions

An ''affine variety'' over an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
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.

## Affine varieties

For an algebraically closed field and a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
, let be an affine -space over , identified to $K^n$ through the choice of an affine coordinate system. The polynomials in the ring can be viewed as ''K''-valued functions on by evaluating at the points in , i.e. by choosing values in ''K'' for each ''xi''. For each set ''S'' of polynomials in , define the zero-locus ''Z''(''S'') to be the set of points in on which the functions in ''S'' simultaneously vanish, that is to say :$Z\left(S\right) = \left \.$ A subset ''V'' of is called an affine algebraic set if ''V'' = ''Z''(''S'') for some ''S''. A nonempty affine algebraic set ''V'' is called irreducible if it cannot be written as the union of two
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
algebraic subsets. An irreducible affine algebraic set is also called an affine variety. (Many authors use the phrase ''affine variety'' to refer to any affine algebraic set, irreducible or not.Hartshorne, p.xv, notes that his choice is not conventional; see for example, Harris, p.3) Affine varieties can be given a natural topology by declaring the
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s to be precisely the affine algebraic sets. This topology is called the Zariski topology. Given a subset ''V'' of , we define ''I''(''V'') to be the ideal of all polynomial functions vanishing on ''V'': :$I\left(V\right) = \left \.$ For any affine algebraic set ''V'', the coordinate ring or structure ring of ''V'' is the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the polynomial ring by this ideal.

## Projective varieties and quasi-projective varieties

Let be an algebraically closed field and let be the projective ''n''-space over . Let in be a homogeneous polynomial of degree ''d''. It is not well-defined to evaluate on points in in homogeneous coordinates. However, because is homogeneous, meaning that , it ''does'' make sense to ask whether vanishes at a point . For each set ''S'' of homogeneous polynomials, define the zero-locus of ''S'' to be the set of points in on which the functions in ''S'' vanish: :$Z\left(S\right) = \.$ A subset ''V'' of is called a projective algebraic set if ''V'' = ''Z''(''S'') for some ''S''. An irreducible projective algebraic set is called a projective variety. Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed. Given a subset ''V'' of , let ''I''(''V'') be the ideal generated by all homogeneous polynomials vanishing on ''V''. For any projective algebraic set ''V'', the
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
of ''V'' is the quotient of the polynomial ring by this ideal. A
quasi-projective variety In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in ...
is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. 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.

## Abstract varieties

In classical algebraic geometry, all varieties were by definition quasi-projective varieties, meaning that they were open subvarieties of closed subvarieties of
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
. For example, in Chapter 1 of Hartshorne a ''variety'' over an algebraically closed field is defined to be a
quasi-projective variety In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in ...
, 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 In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
. 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 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 product is not a variety until it is embedded into the projective space; this is usually done by the Segre embedding. However, any variety that admits one embedding into projective space admits many others by composing the embedding with the Veronese embedding. Consequently, many notions that should be intrinsic, such as the concept of a regular function, are not obviously so. The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made by
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. ...
. In his '' Foundations of Algebraic Geometry'', Weil defined an abstract algebraic variety using valuations. Claude Chevalley made a definition of a scheme, 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 an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
, separated scheme of finite type over an algebraically closed field, although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed.Liu, Qing. ''Algebraic Geometry and Arithmetic Curves'', p. 55 Definition 2.3.47, and p. 88 Example 3.2.3 Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field.

### Existence of non-quasiprojective abstract algebraic varieties

One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata. Nagata's example was not complete (the analog of compactness), but soon afterwards he found an algebraic surface that was complete and non-projective. Since then other examples have been found; for example, it is straightforward to construct a toric variety that is not quasi-projective but complete.

# Examples

## Subvariety

A subvariety is a subset of a variety that is itself a variety (with respect to the structure induced from 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 homogeneous prime ideals of the coordinate ring of the variety.

## Affine variety

### Example 1

Let , and A2 be the two-dimensional
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
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 : :$f\left(x, y\right) = x+y-1.$ The zero-locus of 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 Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
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 : :$Z\left(f\right) = \.$ Thus the subset 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 , 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''): :$g\left(x, y\right) = x^2 + y^2 - 1.$ 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 ''x''2 + ''y''2 = 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 In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
; this name is also often given to the whole variety.

### Example 3

The following example is neither a
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclide ...
, nor a
linear space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
, nor a single point. Let A3 be the three-dimensional affine space over C. The set of points (''x'', ''x''2, ''x''3) for ''x'' in C is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane.Harris, p.9; that it is irreducible is stated as an exercise in Hartshorne p.7 It is the
twisted cubic In mathematics, a twisted cubic is a smooth, rational curve ''C'' of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (''the'' twisted cubic, therefore ...
shown in the above figure. It may be defined by the equations :$\begin y-x^2&=0\\ z-x^3&=0 \end$ 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 In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
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 In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Grö ...
computation to compute the dimension, followed by a random linear change of variables (not always needed); then a
Gröbner basis In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Grö ...
computation for another
monomial order In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all ( monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., * If u \leq v an ...
ing to compute the projection and to prove that it is generically injective and that its image is a
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclide ...
, and finally a
polynomial factorization In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same dom ...
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 ''n''2-space $\mathbb^$ with coordinates $x_$ such that $x_\left(A\right)$ is the (''i'', ''j'')-th entry of the matrix $A$. The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
$\det$ is then a polynomial in $x_$ and thus defines the hypersurface $H = V\left(\det\right)$ in $\mathbb^$. The complement of $H$ is then an open subset of $\mathbb^$ that consists of all the invertible ''n''-by-''n'' matrices, the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
$\operatorname_n\left(k\right)$. It is an affine variety, since, in general, the complement of a hypersurface in an affine variety is affine. Explicitly, consider $\mathbb^ \times \mathbb^1$ where the affine line is given coordinate ''t''. Then $\operatorname_n\left(k\right)$ amounts to the zero-locus in $\mathbb^ \times \mathbb^1$ of the polynomial in $x_, t$: : i.e., the set of matrices ''A'' such that $t \det\left(A\right) = 1$ has a solution. This is best seen algebraically: the coordinate ring of $\operatorname_n\left(k\right)$ is the localization , which can be identified with . The multiplicative group k* of the base field ''k'' is the same as $\operatorname_1\left(k\right)$ and thus is an affine variety. A finite product of it $\left(k^*\right)^r$ 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.

## Projective variety

A
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
is a closed subvariety of a projective space. That is, it is the zero locus of a set of
homogeneous polynomials In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
that generate a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
.

### Example 1

A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. The
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
P1 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 P2 called an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
. 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 ''Gn''(''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 In mathematics, the Plücker map embeds the Grassmannian \mathbf(k,V), whose elements are ''k''- dimensional subspaces of an ''n''-dimensional vector space ''V'', in a projective space, thereby realizing it as an algebraic variety. More precise ...
: : where ''bi'' are any set of linearly independent vectors in ''V'', $\wedge^n V$ is the ''n''-th
exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
of ''V'', and the bracket 'w''means the line spanned by the nonzero vector ''w''. The Grassmannian variety comes with a natural
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every ...
(or
locally free sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
in other terminology) called the tautological bundle, which is important in the study of characteristic classes such as Chern classes.

### Jacobian variety

Let ''C'' be a smooth complete curve and $\operatorname\left(C\right)$ the
Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a globa ...
of it; i.e., the group of isomorphism classes of line bundles on ''C''. Since ''C'' is smooth, $\operatorname\left(C\right)$ can be identified as the divisor class group of ''C'' and thus there is the degree homomorphism $\operatorname : \operatorname\left(C\right) \to \mathbb$. The
Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian var ...
$\operatorname\left(C\right)$ 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 In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
structure on it (the name "abelian" is however not because it is an abelian group). An abelian variety turns out to be projective (
theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field the ...
s in the algebraic setting gives an embedding); thus, $\operatorname\left(C\right)$ is a projective variety. The tangent space to $\operatorname\left(C\right)$ at the identity element is naturally isomorphic to $\operatorname^1\left(C, \mathcal_C\right);$ hence, the dimension of $\operatorname\left(C\right)$ is the genus of $C$. Fix a point $P_0$ on $C$. For each integer $n > 0$, there is a natural morphism :

### Moduli varieties

Given an integer $g \ge 0$, the set of isomorphism classes of smooth complete curves of genus $g$ is called the moduli of curves of genus $g$ and is denoted as $\mathfrak_g$. 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 $g \ge 2$, a not-necessarily-smooth complete curve with no terribly bad singularities and not-so-large automorphism group. The moduli of stable curves $\overline_g$, the set of isomorphism classes of stable curves of genus $g \ge 2$, is then a projective variety which contains $\mathfrak_g$ as an open subset. Since $\overline_g$ is obtained by adding boundary points to $\mathfrak_g$, $\overline_g$ is colloquially said to be a compactification of $\mathfrak_g$. Historically a paper of Mumford and Deligne introduced the notion of a stable curve to show $\mathfrak_g$ is irreducible when $g \ge 2$. 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 A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
and semistable vector bundles on a smooth complete curve $C$. The moduli of semistable vector bundles of a given rank $n$ and a given degree $d$ (degree of the determinant of the bundle) is then a projective variety denoted as $SU_C\left(n, d\right)$, which contains the set $U_C\left(n, d\right)$ of isomorphism classes of stable vector bundles of rank $n$ and degree $d$ as an open subset. Since a line bundle is stable, such a moduli is a generalization of the Jacobian variety of $C$. 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 $\mathbb$ is the problem of compactifying $D / \Gamma$, the quotient of a bounded symmetric domain $D$ by an action of an arithmetic discrete group $\Gamma$. A basic example of $D / \Gamma$ is when $D = \mathfrak_g$, Siegel's upper half-space and $\Gamma$ commensurable with $\operatorname\left(2g, \mathbb\right)$; in that case, $D / \Gamma$ has an interpretation as the moduli $\mathfrak_g$ of principally polarized complex abelian varieties of dimension $g$ (a principal polarization identifies an abelian variety with its dual). The theory of toric varieties (or torus embeddings) gives a way to compactify $D / \Gamma$, a toroidal compactification of it. But there are other ways to compactify $D / \Gamma$; for example, there is the minimal compactification of $D / \Gamma$ due to Baily and Borel: it is the projective variety associated to the graded ring formed by
modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory o ...
s (in the Siegel case, Siegel modular forms). 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. It is an algebraic variety since it is a product of varieties. It is not affine since P1 is a closed subvariety of ''X'' (as the zero locus of ''p''), 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 ''X''; namely, ''p''. Another example of a non-affine non-projective variety is (cf. '.)

## Non-examples

Consider the affine line $\mathbb^1$ over $\mathbb$. The complement of the circle $\$ in $\mathbb^1 = \mathbb$ is not an algebraic variety (not even algebraic set). Note that $, z, ^2 - 1$ is not a polynomial in $z$ (although a polynomial in real variables $x, y$.) On the other hand, the complement of the origin in $\mathbb^1 = \mathbb$ is an algebraic (affine) variety, since the origin is the zero-locus of $z$. 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, a unitary group (over the complex numbers) is not an algebraic variety, while the special linear group $\operatorname_n\left(\mathbb\right)$ is a closed subvariety of $\operatorname_n\left(\mathbb\right)$, the zero-locus of $\det - 1$. (Over a different base field, a unitary group can however be given a structure of a variety.)

# Basic results

* An affine algebraic set ''V'' is a variety
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
''I''(''V'') is a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
; equivalently, ''V'' is a variety if and only if its coordinate ring is an * 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). * The dimension of a variety may be defined in various equivalent ways. See
Dimension of an algebraic variety In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutat ...
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 and a finite product of projective varieties is projective.

# Isomorphism of algebraic varieties

Let be algebraic varieties. We say and are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, and write , if there are regular maps and such that the compositions and are the identity maps on and respectively.

# Discussion and generalizations

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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, becau ...
— 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; 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 In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf o ...
such that every point has a neighbourhood that, as a locally ringed space, is isomorphic to a
spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
. Basically, a variety over is a scheme whose
structure sheaf In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
is a sheaf of -algebras with the property that the rings ''R'' that occur above are all
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
s and are all finitely generated -algebras, that is to say, they are quotients of polynomial algebras by
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
s. This definition works over any field . 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 be ''separated''. (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 having
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
affine charts, and when speaking of a variety only require that the affine charts have trivial nilradical. A complete variety is a variety such that any map from an open subset of a nonsingular
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
into it can be extended uniquely to the whole curve. Every projective variety is complete, but not vice versa. These varieties have been called "varieties in the sense of Serre", since Serre's foundational paper FAC on sheaf cohomology was written for them. They remain typical objects to start studying in algebraic geometry, even if more general objects are also used in an auxiliary way. One way that leads to generalizations is to allow reducible algebraic sets (and fields that aren't algebraically closed), so the rings ''R'' may not be integral domains. A more significant modification is to allow
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
s in the sheaf of rings, that is, rings which are not reduced. This is one of several generalizations of classical algebraic geometry that are built into Grothendieck'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 by ''x''2 = 0 is different from the subscheme defined by ''x'' = 0 (the origin). More generally, the
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
of a morphism of schemes ''X'' → ''Y'' at a point of ''Y'' may be non-reduced, even if ''X'' and ''Y'' are reduced. Geometrically, this says that fibers of good mappings may have nontrivial "infinitesimal" structure. There are further generalizations called algebraic spaces and stacks.

# Algebraic manifolds

An algebraic manifold is an algebraic variety that is also an ''m''-dimensional manifold, and hence every sufficiently small local patch is isomorphic to ''km''. Equivalently, the variety is smooth (free from singular points). When is the real numbers, R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. Projective algebraic manifolds are an equivalent definition for projective varieties. The
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
is one example.

*
Variety (disambiguation) Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film) ...
— listing also several mathematical meanings * Function field of an algebraic variety *
Birational geometry In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational ...
* Abelian variety * Motive (algebraic geometry) * Analytic variety *
Zariski–Riemann space In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring ''k'' of a field ''K'' is a locally ringed space whose points are valuation rings containing ''k'' and contained in ''K''. They generalize the Riemann surface of a comp ...
* Semi-algebraic set

# References

## Sources

* * * * Milne J.
Jacobian Varieties
published as Chapter VII of Arithmetic geometry (Storrs, Conn., 1984), 167–212, Springer, New York, 1986. * * {{Authority control Algebraic geometry