Algebraic varieties are the central objects of study in
algebraic geometry, 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 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 fo ...
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
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
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 establishes a link between
algebra 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 (an algebraic object) in one variable with complex number coefficients is determined by the set of its
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
(a geometric object) in the
complex plane. Generalizing this result,
Hilbert's Nullstellensatz provides a fundamental correspondence between
ideals of
polynomial rings and algebraic sets. Using the ''Nullstellensatz'' and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of
ring theory. This correspondence is a defining feature of algebraic geometry.
Many algebraic varieties are
manifolds, 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 coord ...
. 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 A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
theory, an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field whose
structure morphism is separated and of finite type.
Overview and definitions
An ''affine variety'' over an
algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way. The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but
Nagata Nagata is a surname which can be either of Japanese (written: 永田 or 長田) or Fijian origin. Notable people with the surname include:
* Akira Nagata (born 1985), Japanese vocalist and actor
* Alipate Nagata, Fijian politician
* Anna Nagata (bo ...
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
through the choice of an
affine coordinate system
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 rela ...
. 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 ''x
i''. 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
:
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 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 sets 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'':
:
For any affine algebraic set ''V'', the coordinate ring or structure ring of ''V'' is the
quotient 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
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; ...
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:
:
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 of ''V'' is the quotient of the polynomial ring by this ideal.
A
quasi-projective variety 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. For example, in Chapter 1 of Hartshorne a ''variety'' over an algebraically closed field is defined to be a
quasi-projective variety, 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. 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 In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after ...
. 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. In his ''
Foundations of Algebraic Geometry'', Weil defined an abstract algebraic variety using
valuations.
Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a fou ...
made a definition of a
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
, 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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
(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 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 :
:
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 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 :
:
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''):
:
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 In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.. For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the intege ...
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'', ''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 shown in the above figure. It may be defined by the equations
:
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 ''n''2-space with coordinates such that is the (''i'', ''j'')-th entry of the matrix . 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 a ...
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 ''n''-by-''n'' 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 ''t''. Then amounts to the zero-locus in of the polynomial in :
:
i.e., the set of matrices ''A'' 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 k* of the base field ''k'' is the same as and thus is an affine variety. A finite product of it is an algebraic torus In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine algebraic group commonly found in Projective scheme, projective algebraic geometry and toric ...
, which is again an affine variety.
A general linear group is an example of a linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I ...
, 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 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 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
:
in the 2-dimensional affine space (over a field of characteristic not two). It has the associated cubic homogeneous polynomial equation:
:
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. I ...
. 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
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on ...
).
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:
:
where ''bi'' are any set of linearly independent vectors in ''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 In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k-dimension (vector space), dimensional linear subspace, subspaces of V, given a point in the Grassmannian ...
, which is important in the study of characteristic classes such as Chern classes.
Jacobian variety
Let ''C'' be a smooth complete curve and the Picard group of it; i.e., the group of isomorphism classes of line bundles on ''C''. Since ''C'' is smooth, can be identified as the divisor class group of ''C'' and thus there is the degree homomorphism . The Jacobian variety 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 functions in the algebraic setting gives an embedding); thus, is a projective variety. The tangent space to at the identity element is naturally isomorphic to hence, the dimension of is the genus of .
Fix a point on . For each integer , there is a natural morphism
: