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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 projective 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 ...
''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of
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 ...
s of ''n'' + 1 variables with coefficients in ''k'', 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 ...
, the defining ideal of the variety. Equivalently, an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
is projective if it can be embedded as a Zariski closed
subvariety A subvariety (Latin: ''subvarietas'') in botanical nomenclature is a taxonomic rank. They are rarely used to classify organisms. Plant taxonomy Subvariety is ranked: *below that of variety (''varietas'') *above that of form (''forma''). Subva ...
of \mathbb^n. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single
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 ...
. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
:k _0, \ldots, x_nI is called the
homogeneous coordinate ring In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X'N' ...
of ''X''. Basic invariants of ''X'' such as the
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
and the
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 ...
can be read off the
Hilbert polynomial In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homoge ...
of this
graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
. Projective varieties arise in many ways. They are
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 ...
, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, but
Chow's lemma Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following: :If X ...
describes the close relation of these two notions. Showing that a variety is projective is done by studying
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
s or
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s on ''X''. A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties,
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexa ...
can be viewed as an analog of
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact a ...
. It also leads to the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
for projective curves, i.e., projective varieties of
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 ...
1. The theory of projective curves is particularly rich, including a classification by the
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
of the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties.
Hilbert scheme In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a d ...
s parametrize closed subschemes of \mathbb^n with prescribed Hilbert polynomial. Hilbert schemes, of which
Grassmannian 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 ...
s are special cases, are also projective schemes in their own right.
Geometric invariant theory In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by Group action (mathematics), group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas ...
offers another approach. The classical approaches include the
Teichmüller space In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
and Chow varieties. A particularly rich theory, reaching back to the classics, is available for complex projective varieties, i.e., when the polynomials defining ''X'' have
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
coefficients. Broadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. For example, the theory of
holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a com ...
s (more generally coherent analytic sheaves) on ''X'' coincide with that of algebraic vector bundles. Chow's theorem says that a subset of projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. The combination of analytic and algebraic methods for complex projective varieties lead to areas such as
Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
.


Variety and scheme structure


Variety structure

Let ''k'' be an algebraically closed field. The basis of the definition of projective varieties is projective space \mathbb^n, which can be defined in different, but equivalent ways: * as the set of all lines through the origin in k^ (i.e., all one-dimensional vector subspaces of k^) * as the set of tuples (x_0, \dots, x_n) \in k^, with x_0, \dots, x_n not all zero, modulo the equivalence relation (x_0, \dots, x_n) \sim \lambda (x_0, \dots, x_n) for any \lambda \in k \setminus \. The equivalence class of such a tuple is denoted by _0: \dots: x_n This equivalence class is the general point of projective space. The numbers x_0, \dots, x_n are referred to as the
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 ...
of the point. A ''projective variety'' is, by definition, a closed subvariety of \mathbb^n, where closed refers to 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 ...
. In general, closed subsets of the Zariski topology are defined to be the common zero-locus of a finite collection of homogeneous polynomial functions. Given a polynomial f \in k _0, \dots, x_n/math>, the condition :f( _0: \dots: x_n = 0 does not make sense for arbitrary polynomials, but only if ''f'' is
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
, i.e., the degrees of all the
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer exponent ...
s (whose sum is ''f'') are the same. In this case, the vanishing of :f(\lambda x_0, \dots, \lambda x_n) = \lambda^ f(x_0, \dots, x_n) is independent of the choice of \lambda \ne 0. Therefore, projective varieties arise from homogeneous
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 ...
s ''I'' of k _0, \dots, x_n/math>, and setting :X = \left\. Moreover, the projective variety ''X'' is an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation axiom. Thus, the local study of ''X'' (e.g., singularity) reduces to that of an affine variety. The explicit structure is as follows. The projective space \mathbb^n is covered by the standard open affine charts :U_i = \, which themselves are affine ''n''-spaces with the coordinate ring :k \left ^_1, \dots, y^_n \right \quad y^_j = x_j/x_i. Say ''i'' = 0 for the notational simplicity and drop the superscript (0). Then X \cap U_0 is a closed subvariety of U_0 \simeq \mathbb^n defined by the ideal of k _1, \dots, y_n/math> generated by :f(1, y_1, \dots, y_n) for all ''f'' in ''I''. Thus, ''X'' is an algebraic variety covered by (''n''+1) open affine charts X \cap U_i. Note that ''X'' is the closure of the affine variety X \cap U_0 in \mathbb^n. Conversely, starting from some closed (affine) variety V \subset U_0 \simeq \mathbb^n, the closure of ''V'' in \mathbb^n is the projective variety called the of ''V''. If I \subset k _1, \dots, y_n/math> defines ''V'', then the defining ideal of this closure is the homogeneous ideal of k _0, \dots, x_n/math> generated by :x_0^ f(x_1/x_0, \dots, x_n/x_0) for all ''f'' in ''I''. For example, if ''V'' is an affine curve given by, say, y^2 = x^3 + ax + b in the affine plane, then its projective completion in the projective plane is given by y^2 z = x^3 + ax z^2 + b z^3.


Projective schemes

For various applications, it is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e., \mathbb^n(k) is a scheme which it is a union of (''n'' + 1) copies of the affine ''n''-space ''kn''. More generally, projective space over a ring ''A'' is the union of the affine schemes :U_i = \operatorname A _0/x_i, \dots, x_n/x_i \quad 0 \le i \le n, in such a way the variables match up as expected. The set of
closed point 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 ...
s of \mathbb^n_k, for algebraically closed fields ''k'', is then the projective space \mathbb^n(k) in the usual sense. An equivalent but streamlined construction is given by the
Proj construction In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functo ...
, which is an analog of the
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 the ...
, denoted "Spec", which defines an
affine scheme 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 the ...
. For example, if ''A'' is a ring, then :\mathbb^n_A = \operatornameA _0, \ldots, x_n If ''R'' is a
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 k _0, \ldots, x_n/math> by a homogeneous ideal ''I'', then the canonical surjection induces the
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 ...
:\operatorname R \hookrightarrow \mathbb^n_k. Compared to projective varieties, the condition that the ideal ''I'' be a prime ideal was dropped. This leads to a much more flexible notion: on the one hand the
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
X = \operatorname R may have multiple
irreducible component In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (for ...
s. Moreover, there may be
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 class ...
functions on ''X''. Closed subschemes of \mathbb^n_k correspond bijectively to the homogeneous ideals ''I'' of k _0, \ldots, x_n/math> that are
saturated Saturation, saturated, unsaturation or unsaturated may refer to: Chemistry * Saturation, a property of organic compounds referring to carbon-carbon bonds ** Saturated and unsaturated compounds **Degree of unsaturation ** Saturated fat or fatty ac ...
; i.e., I : (x_0, \dots, x_n) = I. This fact may be considered as a refined version of
projective 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 ...
. We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector space ''V'' over ''k'', we let :\mathbb(V) = \operatorname k /math> where k = \operatorname(V^*) is the
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
of V^*. It is the
projectivization In mathematics, projectivization is a procedure which associates with a non-zero vector space ''V'' a projective space (V), whose elements are one-dimensional subspaces of ''V''. More generally, any subset ''S'' of ''V'' closed under scalar multi ...
of ''V''; i.e., it parametrizes lines in ''V''. There is a canonical surjective map \pi: V \setminus \ \to \mathbb(V), which is defined using the chart described above. One important use of the construction is this (cf., ). A divisor ''D'' on a projective variety ''X'' corresponds to a line bundle ''L''. One then set :, D, = \mathbb(\Gamma(X, L)); it is called the
complete linear system In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the fo ...
of ''D''. Projective space over any
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 ...
''S'' can be defined as a
fiber product of schemes In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determin ...
:\mathbb^n_S = \mathbb_\Z^n \times_ S. If \mathcal(1) is the
twisting sheaf of Serre In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not fun ...
on \mathbb_\Z^n, we let \mathcal(1) denote the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: in ...
of \mathcal(1) to \mathbb^n_S; that is, \mathcal(1) = g^*(\mathcal(1)) for the canonical map g: \mathbb^n_ \to \mathbb^n_. A scheme ''X'' → ''S'' is called projective over ''S'' if it factors as a closed immersion :X \to \mathbb^n_S followed by the projection to ''S''. A line bundle (or invertible sheaf) \mathcal on a scheme ''X'' over ''S'' is said to be very ample relative to ''S'' if there is an
immersion Immersion may refer to: The arts * "Immersion", a 2012 story by Aliette de Bodard * ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux * Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
(i.e., an open immersion followed by a closed immersion) :i: X \to \mathbb^n_S for some ''n'' so that \mathcal(1) pullbacks to \mathcal. Then a ''S''-scheme ''X'' is projective if and only if it is
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 ...
and there exists a very ample sheaf on ''X'' relative to ''S''. Indeed, if ''X'' is proper, then an immersion corresponding to the very ample line bundle is necessarily closed. Conversely, if ''X'' is projective, then the pullback of \mathcal(1) under the closed immersion of ''X'' into a projective space is very ample. That "projective" implies "proper" is deeper: the ''
main theorem of elimination theory In algebraic geometry, the main theorem of elimination theory states that every projective scheme is proper scheme, proper. A version of this theorem predates the existence of scheme theory. It can be stated, proved, and applied in the following mo ...
''.


Relation to complete varieties

By definition, a variety is
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 ...
, if it is
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 ...
over ''k''. The
valuative criterion of properness In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ...
expresses the intuition that in a proper variety, there are no points "missing". There is a close relation between complete and projective varieties: on the one hand, projective space and therefore any projective variety is complete. The converse is not true in general. However: *A
smooth 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 a ...
''C'' is projective if and only if it is
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 ...
. This is proved by identifying ''C'' with the set of
discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' ...
s of the function field ''k''(''C'') over ''k''. This set has a natural Zariski topology called the
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 c ...
. *
Chow's lemma Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following: :If X ...
states that for any complete variety ''X'', there is a projective variety ''Z'' and a
birational morphism 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 fu ...
''Z'' → ''X''. (Moreover, through
normalization Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
, one can assume this projective variety is normal.) Some properties of a projective variety follow from completeness. For example, :\Gamma(X, \mathcal_X) = k for any projective variety ''X'' over ''k''. This fact is an algebraic analogue of Liouville's theorem (any holomorphic function on a connected compact complex manifold is constant). In fact, the similarity between complex analytic geometry and algebraic geometry on complex projective varieties goes much further than this, as is explained below. Quasi-projective varieties are, by definition, those which are open subvarieties of projective varieties. This class of varieties includes
affine varieties 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 ideal. ...
. Affine varieties are almost never complete (or projective). In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globally
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 a projective variety.


Examples and basic invariants

By definition, any homogeneous ideal in a polynomial ring yields a projective scheme (required to be prime ideal to give a variety). In this sense, examples of projective varieties abound. The following list mentions various classes of projective varieties which are noteworthy since they have been studied particularly intensely. The important class of complex projective varieties, i.e., the case k=\Complex, is discussed further below. The product of two projective spaces is projective. In fact, there is the explicit immersion (called
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 ...
) :\begin \mathbb^n \times \mathbb^m \to \mathbb^ \\ (x_i, y_j) \mapsto x_i y_j \end As a consequence, the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of projective varieties over ''k'' is again projective. 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 ...
exhibits a
Grassmannian 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 ...
as a projective variety.
Flag varieties In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flag (linear algebra), flags in a finite-dimensional vector space ''V'' over a field (mathematics), field F. When F is the real or complex nu ...
such as the quotient of 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, ...
\mathrm_n(k) modulo the subgroup of upper triangular matrices, are also projective, which is an important fact in the theory of
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s.


Homogeneous coordinate ring and Hilbert polynomial

As the prime ideal ''P'' defining a projective variety ''X'' is homogeneous, the
homogeneous coordinate ring In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X'N' ...
:R = k _0, \dots, x_n/ P is a
graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
, i.e., can be expressed as the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of its graded components: :R = \bigoplus_ R_n. There exists a polynomial ''P'' such that \dim R_n = P(n) for all sufficiently large ''n''; it is called the
Hilbert polynomial In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homoge ...
of ''X''. It is a numerical invariant encoding some extrinsic geometry of ''X''. The degree of ''P'' is the
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 ...
''r'' of ''X'' and its leading coefficient times r! is the
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
of the variety ''X''. The
arithmetic genus In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface. Projective varieties Let ''X'' be a projective scheme of dimension ''r'' over a field '' ...
of ''X'' is (−1)''r'' (''P''(0) − 1) when ''X'' is smooth. For example, the homogeneous coordinate ring of \mathbb^n is k _0, \ldots, x_n/math> and its Hilbert polynomial is P(z) = \binom; its arithmetic genus is zero. If the homogeneous coordinate ring ''R'' is an
integrally closed domain In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root ...
, then the projective variety ''X'' is said to be
projectively normal In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X'N'' ...
. Note, unlike normality, projective normality depends on ''R'', the embedding of ''X'' into a projective space. The normalization of a projective variety is projective; in fact, it's the Proj of the integral closure of some homogeneous coordinate ring of ''X''.


Degree

Let X \subset \mathbb^N be a projective variety. There are at least two equivalent ways to define the degree of ''X'' relative to its embedding. The first way is to define it as the cardinality of the finite set :\# (X \cap H_1 \cap \cdots \cap H_d) where ''d'' is the dimension of ''X'' and ''H''''i'''s are hyperplanes in "general positions". This definition corresponds to an intuitive idea of a degree. Indeed, if ''X'' is a hypersurface, then the degree of ''X'' is the degree of the homogeneous polynomial defining ''X''. The "general positions" can be made precise, for example, by
intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...
; one requires that the intersection is
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 ...
and that the multiplicities of irreducible components are all one. The other definition, which is mentioned in the previous section, is that the degree of ''X'' is the leading coefficient of the
Hilbert polynomial In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homoge ...
of ''X'' times (dim ''X'')!. Geometrically, this definition means that the degree of ''X'' is the multiplicity of the vertex of the affine cone over ''X''. Let V_1, \dots, V_r \subset \mathbb^N be closed subschemes of pure dimensions that intersect properly (they are in general position). If ''mi'' denotes the multiplicity of an irreducible component ''Zi'' in the intersection (i.e.,
intersection multiplicity In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ta ...
), then the generalization of
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the degr ...
says: :\sum_1^s m_i \deg Z_i = \prod_1^r \deg V_i. The intersection multiplicity ''mi'' can be defined as the coefficient of ''Zi'' in the intersection product V_1 \cdot \cdots \cdot V_r in the
Chow ring In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-c ...
of \mathbb^N. In particular, if H \subset \mathbb^N is a hypersurface not containing ''X'', then :\sum_1^s m_i \deg Z_i = \deg(X) \deg(H) where ''Zi'' are the irreducible components of the
scheme-theoretic intersection In algebraic geometry, the scheme-theoretic intersection of closed subschemes ''X'', ''Y'' of a scheme ''W'' is X \times_W Y, the fiber product of the closed immersions X \hookrightarrow W, Y \hookrightarrow W. It is denoted by X \cap Y. Locally, ' ...
of ''X'' and ''H'' with multiplicity (length of the local ring) ''mi''. A complex projective variety can be viewed as a
compact complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a c ...
; the degree of the variety (relative to the embedding) is then the volume of the variety as a manifold with respect to the metric inherited from the ambient
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
. A complex projective variety can be characterized as a minimizer of the volume (in a sense).


The ring of sections

Let ''X'' be a projective variety and ''L'' a line bundle on it. Then the graded ring :R(X, L) = \bigoplus_^ H^0(X, L^) is called the
ring of sections This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...
of ''L''. If ''L'' is
ample In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of a ...
, then Proj of this ring is ''X''. Moreover, if ''X'' is normal and ''L'' is very ample, then R(X,L) is the integral closure of the homogeneous coordinate ring of ''X'' determined by ''L''; i.e., X \hookrightarrow \mathbb^N so that \mathcal_(1) pulls-back to ''L''. For applications, it is useful to allow for
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s (or \Q-divisors) not just line bundles; assuming ''X'' is normal, the resulting ring is then called a generalized ring of sections. If K_X is a
canonical divisor In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...
on ''X'', then the generalized ring of sections :R(X, K_X) is called the
canonical ring In mathematics, the pluricanonical ring of an algebraic variety ''V'' (which is non-singular), or of a complex manifold, is the graded ring :R(V,K)=R(V,K_V) \, of sections of powers of the canonical bundle ''K''. Its ''n''th graded component (for ...
of ''X''. If the canonical ring is finitely generated, then Proj of the ring is called the
canonical model A canonical model is a design pattern used to communicate between different data formats. Essentially: create a data model which is a superset of all the others ("canonical"), and create a "translator" module or layer to/from which all existing ...
of ''X''. The canonical ring or model can then be used to define the
Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
of ''X''.


Projective curves

Projective schemes of dimension one are called ''projective curves''. Much of the theory of projective curves is about smooth projective curves, since the singularities of curves can be resolved by
normalization Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
, which consists in taking locally the
integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is ...
of the ring of regular functions. Smooth projective curves are isomorphic if and only if their function fields are isomorphic. The study of finite extensions of :\mathbb F_p(t), or equivalently smooth projective curves over \mathbb F_p is an important branch in
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
. A smooth projective curve of genus one is 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 ...
. As a consequence of the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
, such a curve can be embedded as a closed subvariety in \mathbb^2. In general, any (smooth) projective curve can be embedded in \mathbb^3 (for a proof, see Secant variety#Examples). Conversely, any smooth closed curve in \mathbb^2 of degree three has genus one by the genus formula and is thus an elliptic curve. A smooth complete curve of genus greater than or equal to two is called a
hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dist ...
if there is a finite morphism C \to \mathbb^1 of degree two.


Projective hypersurfaces

Every irreducible closed subset of \mathbb^n of codimension one 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 ...
; i.e., the zero set of some homogeneous irreducible polynomial.


Abelian varieties

Another important invariant of a projective variety ''X'' is 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 ...
\operatorname(X) of ''X'', the set of isomorphism classes of line bundles on ''X''. It is isomorphic to H^1(X, \mathcal O_X^*) and therefore an intrinsic notion (independent of embedding). For example, the Picard group of \mathbb^n is isomorphic to \Z via the degree map. The kernel of \deg: \operatorname(X) \to \Z is not only an abstract abelian group, but there is a variety called 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 ''X'', Jac(''X''), whose points equal this group. The Jacobian of a (smooth) curve plays an important role in the study of the curve. For example, the Jacobian of an elliptic curve ''E'' is ''E'' itself. For a curve ''X'' of genus ''g'', Jac(''X'') has dimension ''g''. Varieties, such as the Jacobian variety, which are complete and have a group structure are known as
abelian varieties 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 function ...
, in honor of
Niels Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
. In marked contrast to
affine 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 ...
s such as GL_n(k), such groups are always commutative, whence the name. Moreover, they admit an ample
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
and are thus projective. On the other hand, an
abelian scheme 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 function ...
may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties and
K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected alg ...
s.


Projections

Let E \subset \mathbb^n be a linear subspace; i.e., E = \ for some linearly independent linear functionals ''si''. Then the projection from ''E'' is the (well-defined) morphism :\begin \phi: \mathbb^n - E \to \mathbb^r \\ x \mapsto _0(x) : \cdots : s_r(x)\end The geometric description of this map is as follows: *We view \mathbb^r \subset \mathbb^n so that it is disjoint from ''E''. Then, for any x \in \mathbb^n \setminus E, \phi(x) = W_x \cap \mathbb^r, where W_x denotes the smallest linear space containing ''E'' and ''x'' (called the
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two top ...
of ''E'' and ''x''.) *\phi^(\) = \, where y_i are the homogeneous coordinates on \mathbb^r. *For any closed subscheme Z \subset \mathbb^n disjoint from ''E'', the restriction \phi: Z \to \mathbb^r is a
finite morphism In algebraic geometry, a finite morphism between two affine varieties 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 polyn ...
. Projections can be used to cut down the dimension in which a projective variety is embedded, up to
finite morphism In algebraic geometry, a finite morphism between two affine varieties 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 polyn ...
s. Start with some projective variety X \subset \mathbb^n. If n > \dim X, the projection from a point not on ''X'' gives \phi: X \to \mathbb^. Moreover, \phi is a finite map to its image. Thus, iterating the procedure, one sees there is a finite map :X \to \mathbb^d, \quad d = \dim X. This result is the projective analog of
Noether's normalization lemma In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field ''k'', and any finitely generated commutative ''k''-algebra ''A'', there exists a non-negativ ...
. (In fact, it yields a geometric proof of the normalization lemma.) The same procedure can be used to show the following slightly more precise result: given a projective variety ''X'' over a perfect field, there is a finite birational morphism from ''X'' to a hypersurface ''H'' in \mathbb^. In particular, if ''X'' is normal, then it is the normalization of ''H''.


Duality and linear system

While a projective ''n''-space \mathbb^n parameterizes the lines in an affine ''n''-space, the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
of it parametrizes the hyperplanes on the projective space, as follows. Fix a field ''k''. By \breve_k^n, we mean a projective ''n''-space :\breve_k^n = \operatorname(k _0, \dots, u_n equipped with the construction: :f \mapsto H_f = \, a hyperplane on \mathbb^n_L where f: \operatorname L \to \breve_k^n is an ''L''-point of \breve_k^n for a field extension ''L'' of ''k'' and \alpha_i = f^*(u_i) \in L. For each ''L'', the construction is a bijection between the set of ''L''-points of \breve_k^n and the set of hyperplanes on \mathbb^n_L. Because of this, the dual projective space \breve_k^n is said to be the
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
of hyperplanes on \mathbb^n_k. A line in \breve_k^n is called a
pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail ...
: it is a family of hyperplanes on \mathbb^n_k parametrized by \mathbb^1_k. If ''V'' is a finite-dimensional vector space over ''k'', then, for the same reason as above, \mathbb(V^*) = \operatorname(\operatorname(V)) is the space of hyperplanes on \mathbb(V). An important case is when ''V'' consists of sections of a line bundle. Namely, let ''X'' be an algebraic variety, ''L'' a line bundle on ''X'' and V \subset \Gamma(X, L) a vector subspace of finite positive dimension. Then there is a map: :\begin \varphi_V: X \setminus B \to \mathbb(V^*) \\ x \mapsto H_x = \ \end determined by the linear system ''V'', where ''B'', called the
base locus In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the fo ...
, is the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of the divisors of zero of nonzero sections in ''V'' (see Linear system of divisors#A map determined by a linear system for the construction of the map).


Cohomology of coherent sheaves

Let ''X'' be a projective scheme over a field (or, more generally over a Noetherian ring ''A''). Cohomology of coherent sheaves \mathcal F on ''X'' satisfies the following important theorems due to Serre: #H^p(X, \mathcal) is a finite-dimensional ''k''-vector space for any ''p''. #There exists an integer n_0 (depending on \mathcal; see also
Castelnuovo–Mumford regularity In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf ''F'' over projective space \mathbf^n is the smallest integer ''r'' such that it is r-regular, meaning that :H^i(\mathbf^n, F(r-i))=0 whenever i>0. The regularity of a ...
) such that H^p(X, \mathcal(n)) = 0 for all n \ge n_0 and ''p'' > 0, where \mathcal F(n) = \mathcal F \otimes \mathcal O(n) is the twisting with a power of a very ample line bundle \mathcal(1). These results are proven reducing to the case X= \mathbb^n using the isomorphism :H^p(X, \mathcal) = H^p(\mathbb^r, \mathcal), p \ge 0 where in the right-hand side \mathcal is viewed as a sheaf on the projective space by extension by zero. The result then follows by a direct computation for \mathcal = \mathcal_(n), ''n'' any integer, and for arbitrary \mathcal F reduces to this case without much difficulty. As a corollary to 1. above, if ''f'' is a projective morphism from a noetherian scheme to a noetherian ring, then the higher direct image R^p f_* \mathcal is coherent. The same result holds for proper morphisms ''f'', as can be shown with the aid of
Chow's lemma Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following: :If X ...
.
Sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...
groups ''Hi'' on a noetherian topological space vanish for ''i'' strictly greater than the dimension of the space. Thus the quantity, called the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
of \mathcal, :\chi(\mathcal) = \sum_^\infty (-1)^i \dim H^i(X, \mathcal) is a well-defined integer (for ''X'' projective). One can then show \chi(\mathcal(n)) = P(n) for some polynomial ''P'' over rational numbers. Applying this procedure to the structure sheaf \mathcal_X, one recovers the Hilbert polynomial of ''X''. In particular, if ''X'' is irreducible and has dimension ''r'', the arithmetic genus of ''X'' is given by :(-1)^r (\chi(\mathcal_X) - 1), which is manifestly intrinsic; i.e., independent of the embedding. The arithmetic genus of a hypersurface of degree ''d'' is \binom in \mathbb^n. In particular, a smooth curve of degree ''d'' in \mathbb^2 has arithmetic genus (d-1)(d-2)/2. This is the genus formula.


Smooth projective varieties

Let ''X'' be a smooth projective variety where all of its irreducible components have dimension ''n''. In this situation, the
canonical sheaf In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, ...
ω''X'', defined as the sheaf of
Kähler differential In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic ge ...
s of top degree (i.e., algebraic ''n''-forms), is a line bundle.


Serre duality

Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexa ...
states that for any locally free sheaf \mathcal on ''X'', :H^i(X, \mathcal) \simeq H^(X, \mathcal^\vee \otimes \omega_X)' where the superscript prime refers to the dual space and \mathcal^\vee is the dual sheaf of \mathcal. A generalization to projective, but not necessarily smooth schemes is known as
Verdier duality In mathematics, Verdier duality is a cohomology, cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact space, locally compact topolo ...
.


Riemann–Roch theorem

For a (smooth projective) curve ''X'', ''H''2 and higher vanish for dimensional reason and the space of the global sections of the structure sheaf is one-dimensional. Thus the arithmetic genus of ''X'' is the dimension of H^1(X, \mathcal_X). By definition, the
geometric genus In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for complex m ...
of ''X'' is the dimension of ''H''0(''X'', ''ω''''X''). Serre duality thus implies that the arithmetic genus and the geometric genus coincide. They will simply be called the genus of ''X''. Serre duality is also a key ingredient in the proof of the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
. Since ''X'' is smooth, there is an isomorphism of groups : \begin \operatorname(X) \to \operatorname(X) \\ D \mapsto \mathcal(D) \end from the group of (Weil) divisors modulo principal divisors to the group of isomorphism classes of line bundles. A divisor corresponding to ω''X'' is called the canonical divisor and is denoted by ''K''. Let ''l''(''D'') be the dimension of H^0(X, \mathcal(D)). Then the Riemann–Roch theorem states: if ''g'' is a genus of ''X'', :l(D) -l(K - D) = \deg D + 1 - g, for any divisor ''D'' on ''X''. By the Serre duality, this is the same as: :\chi(\mathcal(D)) = \deg D + 1 - g, which can be readily proved. A generalization of the Riemann–Roch theorem to higher dimension is the
Hirzebruch–Riemann–Roch theorem In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex alge ...
, as well as the far-reaching
Grothendieck–Riemann–Roch theorem In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is it ...
.


Hilbert schemes

''
Hilbert scheme In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a d ...
s'' parametrize all closed subvarieties of a projective scheme ''X'' in the sense that the points (in the functorial sense) of ''H'' correspond to the closed subschemes of ''X''. As such, the Hilbert scheme is an example of a
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
, i.e., a geometric object whose points parametrize other geometric objects. More precisely, the Hilbert scheme parametrizes closed subvarieties whose
Hilbert polynomial In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homoge ...
equals a prescribed polynomial ''P''. It is a deep theorem of Grothendieck that there is a scheme H_X^P over ''k'' such that, for any ''k''-scheme ''T'', there is a bijection :\ \ \ \longleftrightarrow \ \ \ The closed subscheme of X \times H_X^P that corresponds to the identity map H_X^P \to H_X^P is called the ''universal family''. For P(z) = \binom, the Hilbert scheme H_^P is called the
Grassmannian 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 ...
of ''r''-planes in \mathbb^n and, if ''X'' is a projective scheme, H_X^P is called the
Fano scheme In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a Complete algebraic variety, complete variety ''X'' whose anticanonical bundle ''K''X* is ample line bundle, ample. In this definition, one could assume that ''X'' is Smooth s ...
of ''r''-planes on ''X''.


Complex projective varieties

In this section, all algebraic varieties are
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
algebraic varieties. A key feature of the theory of complex projective varieties is the combination of algebraic and analytic methods. The transition between these theories is provided by the following link: since any complex polynomial is also a holomorphic function, any complex variety ''X'' yields a complex
analytic space An analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study of several complex variables, but they also a ...
, denoted X(\Complex). Moreover, geometric properties of ''X'' are reflected by the ones of X(\Complex). For example, the latter is a
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
if and only if ''X'' is smooth; it is compact if and only if ''X'' is proper over \Complex.


Relation to complex Kähler manifolds

Complex projective space is a
Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
. This implies that, for any projective algebraic variety ''X'', X(\Complex) is a compact Kähler manifold. The converse is not in general true, but the
Kodaira embedding theorem In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials. ...
gives a criterion for a Kähler manifold to be projective. In low dimensions, there are the following results: *(Riemann) A
compact Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
(i.e., compact complex manifold of dimension one) is a projective variety. By the
Torelli theorem In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) ''C'' is determined b ...
, it is uniquely determined by its Jacobian. *(Chow-Kodaira) A compact
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
of dimension two with two algebraically independent
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
s is a projective variety.


GAGA and Chow's theorem

Chow's theorem provides a striking way to go the other way, from analytic to algebraic geometry. It states that every analytic subvariety of a complex projective space is algebraic. The theorem may be interpreted to saying that a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
satisfying certain growth condition is necessarily algebraic: "projective" provides this growth condition. One can deduce from the theorem the following: * Meromorphic functions on the complex projective space are rational. * If an algebraic map between algebraic varieties is an analytic
isomorphism 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 is ...
, then it is an (algebraic) isomorphism. (This part is a basic fact in complex analysis.) In particular, Chow's theorem implies that a holomorphic map between projective varieties is algebraic. (consider the graph of such a map.) * Every
holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a com ...
on a projective variety is induced by a unique algebraic vector bundle. * Every holomorphic line bundle on a projective variety is a line bundle of a divisor. Chow's theorem can be shown via Serre's GAGA principle. Its main theorem states: :Let ''X'' be a projective scheme over \Complex. Then the functor associating the coherent sheaves on ''X'' to the coherent sheaves on the corresponding complex analytic space ''X''an is an equivalence of categories. Furthermore, the natural maps ::H^i(X, \mathcal) \to H^i(X^\text, \mathcal) :are isomorphisms for all ''i'' and all coherent sheaves \mathcal on ''X''.


Complex tori vs. complex abelian varieties

The complex manifold associated to an abelian variety ''A'' over \Complex is a compact
complex Lie group In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mat ...
. These can be shown to be of the form :\Complex^g / L and are also referred to as
complex tori In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
. Here, ''g'' is the dimension of the torus and ''L'' is a lattice (also referred to as
period lattice In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. Definition ...
). According to the
uniformization theorem In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization o ...
already mentioned above, any torus of dimension 1 arises from an abelian variety of dimension 1, i.e., from 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 ...
. In fact, the
Weierstrass's elliptic function In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by t ...
\wp attached to ''L'' satisfies a certain differential equation and as a consequence it defines a closed immersion: :\begin \Complex/L \to \mathbb^2 \\ L \mapsto (0:0:1) \\ z \mapsto (1 : \wp(z) : \wp'(z)) \end There is a ''p''-adic analog, the
p-adic uniformization In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensio ...
theorem. For higher dimensions, the notions of complex abelian varieties and complex tori differ: only polarized complex tori come from abelian varieties.


Kodaira vanishing

The fundamental
Kodaira vanishing theorem In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices ''q'' > 0 are automatically zero. The implica ...
states that for an ample line bundle \mathcal on a smooth projective variety ''X'' over a field of characteristic zero, :H^i(X, \mathcal\otimes \omega_X) = 0 for ''i'' > 0, or, equivalently by Serre duality H^i(X, \mathcal L^) = 0 for ''i'' < ''n''. The first proof of this theorem used analytic methods of Kähler geometry, but a purely algebraic proof was found later. The Kodaira vanishing in general fails for a smooth projective variety in positive characteristic. Kodaira's theorem is one of various vanishing theorems, which give criteria for higher sheaf cohomologies to vanish. Since the Euler characteristic of a sheaf (see above) is often more manageable than individual cohomology groups, this often has important consequences about the geometry of projective varieties.


Related notions

* Multi-projective variety * ''Weighted projective variety'', a closed subvariety of a
weighted projective space In algebraic geometry, a weighted projective space P(''a''0,...,''a'n'') is the projective variety Proj(''k'' 'x''0,...,''x'n'' associated to the graded ring ''k'' 'x''0,...,''x'n''where the variable ''x'k'' has degree ''a'k''. Prop ...


See also

*
Algebraic geometry of projective spaces Projective space plays a central role in algebraic geometry. The aim of this article is to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective space. Homogeneous polynomial ideals Let k be an al ...
*
Adequate equivalence relation In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined i ...
*
Hilbert scheme In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a d ...
*
Lefschetz hyperplane theorem In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the ...
*
Minimal model program In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its orig ...


Notes


References

* * * * * * * * * * * * * *R. Vakil
Foundations Of Algebraic Geometry
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External links


The Hilbert Scheme
by Charles Siegel - a blog post
varieties Ch. 1
Algebraic geometry Algebraic varieties Projective geometry