![Twisted cubic curve](https://upload.wikimedia.org/wikipedia/commons/8/89/Twisted_cubic_curve.png)
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 s ...
over the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
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
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of two smaller
sets that are
closed in the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
. 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 polynomial ...
establishes a link between
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 a ...
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^+\cd ...
(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
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 the ...
. Generalizing this result,
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
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 variables) ...
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 n ...
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 Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
. Algebraic varieties of dimension one are called
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s and algebraic varieties of dimension two are called
algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s.
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
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 ...
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 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 (bor ...
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 n ...
, let be an
affine -space over , identified to
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 ''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
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
In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that ...
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 cl ...
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'':
:
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
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; t ...
of degree ''d''. It is not well-defined to evaluate on points in in
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
. 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
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 s ...
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 s ...
, 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 function In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular ...
s 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 In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre.
Definition
The Segre map may be defined as the map ...
. 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
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. Th ...
. In his ''
Foundations of Algebraic Geometry
''Foundations of Algebraic Geometry'' is a book by that develops algebraic geometry over field (mathematics), fields of any characteristic (algebra), characteristic. In particular it gives a careful treatment of intersection theory by defining th ...
'', 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 foundin ...
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
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 i ...
,
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 In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be nor ...
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
In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formaliz ...
.
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
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 relate ...
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
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:
Arts ...
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
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a dat ...
. 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
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 Eucl ...
; 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 Euclidean ...
, 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 can ...
, 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
:
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 contrapositiv ...
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 Gröbn ...
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 Gröbn ...
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 and ...
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 Euclidean ...
, 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 (mathematics), field or in the integers as the product of irreducible polynomial, irreducible ...
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 and ...
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
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, ...
. 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
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
, 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 algebraic geometry and toric geometry. Higher ...
, 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_n wh ...
, an affine variety that has a structure of a group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
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 w ...
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 with ...
.
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
:
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. 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
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
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
''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 precisely ...
:
:
where ''bi'' are any set of linearly independent vectors in ''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 po ...
(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 In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k-dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector su ...
, which is important in the study of characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
es such as Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
es.
Jacobian variety
Let ''C'' be a smooth complete curve and 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 global ve ...
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
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumfo ...
of ''C'' and thus there is the degree homomorphism . 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 vari ...
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
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
, 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 commut ...
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 theo ...
s 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
: