Projective variety

In algebraic geometry, a projective variety over an algebraically closed field "k" is a subset of some projective "n"-space formula_1 over "k" that is the zero-locus of some finite family of homogeneous polynomials of "n" + 1 variables with coefficients in "k", that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of formula_1.
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.
If "X" is a projective variety defined by a homogeneous prime ideal "I", then the quotient ring
is called the homogeneous coordinate ring of "X". Basic invariants of "X" such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring.
Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, but Chow's lemma describes the close relation of these two notions. Showing that a variety is projective is done by studying line bundles or divisors on "X".
A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties, Serre duality can be viewed as an analog of Poincaré duality. It also leads to the Riemann-Roch theorem for projective curves, i.e., projective varieties of dimension 1. The theory of projective curves is particularly rich, including a classification by the genus of the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties. Hilbert schemes parametrize closed subschemes of formula_1 with prescribed Hilbert polynomial. Hilbert schemes, of which Grassmannians are special cases, are also projective schemes in their own right. Geometric invariant theory offers another approach. The classical approaches include the Teichmüller space 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 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 bundles (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.
Variety and scheme structure.
Variety structure.
Let "k" be an algebraically closed field. The basis of the definition of projective varieties is projective space formula_1, which can be defined in different, but equivalent ways:
A "projective variety" is, by definition, a closed subvariety of formula_1, where closed refers to the Zariski topology. 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 formula_15, the condition
does not make sense for arbitrary polynomials, but only if "f" is homogeneous, i.e., the degrees of all the monomials (whose sum is "f") are the same. In this case, the vanishing of
is independent of the choice of formula_18.
Therefore, projective varieties arise from homogeneous prime ideals "I" of formula_19, and setting
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 formula_1 is covered by the standard open affine charts
which themselves are affine "n"-spaces with the coordinate ring
Say "i" = 0 for the notational simplicity and drop the superscript (0). Then formula_24 is a closed subvariety of formula_25 defined by the ideal of formula_26 generated by
for all "f" in "I". Thus, "X" is an algebraic variety covered by ("n"+1) open affine charts formula_28.
Note that "X" is the closure of the affine variety formula_24 in formula_1. Conversely, starting from some closed (affine) variety formula_31, the closure of "V" in formula_1 is the projective variety called the of "V". If formula_33 defines "V", then the defining ideal of this closure is the homogeneous ideal of formula_19 generated by
for all "f" in "I".
For example, if "V" is an affine curve given by, say, formula_36 in the affine plane, then its projective completion in the projective plane is given by formula_37
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., formula_38 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
in such a way the variables match up as expected. The set of closed points of formula_40, for algebraically closed fields "k", is then the projective space formula_38 in the usual sense.
An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted "Spec", which defines an affine scheme. For example, if "A" is a ring, then
If "R" is a quotient of formula_43 by a homogeneous ideal "I", then the canonical surjection induces the closed immersion
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 formula_45 may have multiple irreducible components. Moreover, there may be nilpotent functions on "X".
Closed subschemes of formula_40 correspond bijectively to the homogeneous ideals "I" of formula_43 that are saturated; i.e., formula_48 This fact may be considered as a refined version of projective Nullstellensatz.
We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector space "V" over "k", we let
where formula_50 is the symmetric algebra of formula_51. It is the projectivization of "V"; i.e., it parametrizes lines in "V". There is a canonical surjective map formula_52, 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
it is called the complete linear system of "D".
Projective space over any scheme "S" can be defined as a fiber product of schemes
If formula_55 is the twisting sheaf of Serre on formula_56, we let formula_57 denote the pullback of formula_55 to formula_59; that is, formula_60 for the canonical map formula_61
A scheme "X" → "S" is called projective over "S" if it factors as a closed immersion
followed by the projection to "S".
A line bundle (or invertible sheaf) formula_63 on a scheme "X" over "S" is said to be very ample relative to "S" if there is an immersion (i.e., an open immersion followed by a closed immersion)
for some "n" so that formula_55 pullbacks to formula_63. Then a "S"-scheme "X" is projective if and only if it is proper 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 formula_55 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".
Relation to complete varieties.
By definition, a variety is complete, if it is proper over "k". The valuative criterion of properness 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:
Some properties of a projective variety follow from completeness. For example,
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. 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 functions 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 formula_69, is discussed further below.
The product of two projective spaces is projective. In fact, there is the explicit immersion (called Segre embedding)
As a consequence, the product of projective varieties over "k" is again projective. The Plücker embedding exhibits a Grassmannian as a projective variety. Flag varieties such as the quotient of the general linear group formula_71 modulo the subgroup of upper triangular matrices, are also projective, which is an important fact in the theory of algebraic groups.
Homogeneous coordinate ring and Hilbert polynomial.
As the prime ideal "P" defining a projective variety "X" is homogeneous, the homogeneous coordinate ring
is a graded ring, i.e., can be expressed as the direct sum of its graded components:
There exists a polynomial "P" such that formula_74 for all sufficiently large "n"; it is called the Hilbert polynomial of "X". It is a numerical invariant encoding some extrinsic geometry of "X". The degree of "P" is the dimension "r" of "X" and its leading coefficient times r! is the degree of the variety "X". The arithmetic genus of "X" is (−1)"r" ("P"(0) − 1) when "X" is smooth.
For example, the homogeneous coordinate ring of formula_1 is formula_43 and its Hilbert polynomial is formula_77; its arithmetic genus is zero.
If the homogeneous coordinate ring "R" is an integrally closed domain, then the projective variety "X" is said to be projectively normal. 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 formula_78 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
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; one requires that the intersection is proper 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 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 formula_80 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), then the generalization of Bézout's theorem says:
The intersection multiplicity "mi" can be defined as the coefficient of "Zi" in the intersection product formula_82 in the Chow ring of formula_83.
In particular, if formula_84 is a hypersurface not containing "X", then
where "Zi" are the irreducible components of the scheme-theoretic intersection of "X" and "H" with multiplicity (length of the local ring) "mi".
A complex projective variety can be viewed as a compact complex manifold; 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. 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
is called the ring of sections of "L". If "L" is ample, then Proj of this ring is "X". Moreover, if "X" is normal and "L" is very ample, then formula_87 is the integral closure of the homogeneous coordinate ring of "X" determined by "L"; i.e., formula_88 so that formula_89 pulls-back to "L".
For applications, it is useful to allow for divisors (or formula_90-divisors) not just line bundles; assuming "X" is normal, the resulting ring is then called a generalized ring of sections. If formula_91 is a canonical divisor on "X", then the generalized ring of sections
is called the canonical ring of "X". If the canonical ring is finitely generated, then Proj of the ring is called the canonical model of "X". The canonical ring or model can then be used to define the Kodaira dimension 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, which consists in taking locally the integral closure 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
or equivalently smooth projective curves over formula_94 is an important branch in algebraic number theory.
A smooth projective curve of genus one is called an elliptic curve. As a consequence of the Riemann-Roch theorem, such a curve can be embedded as a closed subvariety in formula_95. In general, any (smooth) projective curve can be embedded in formula_96 (for a proof, see Secant variety#Examples). Conversely, any smooth closed curve in formula_95 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 if there is a finite morphism formula_98 of degree two.
Projective hypersurfaces.
Every irreducible closed subset of formula_1 of codimension one is a hypersurface; i.e., the zero set of some homogeneous irreducible polynomial.
Abelian varieties.
Another important invariant of a projective variety "X" is the Picard group formula_100 of "X", the set of isomorphism classes of line bundles on "X". It is isomorphic to formula_101 and therefore an intrinsic notion (independent of embedding). For example, the Picard group of formula_1 is isomorphic to formula_103 via the degree map. The kernel of formula_104 is not only an abstract abelian group, but there is a variety called the Jacobian variety 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 honor of Niels Abel. In marked contrast to affine algebraic groups such as formula_105, such groups are always commutative, whence the name. Moreover, they admit an ample line bundle and are thus projective. On the other hand, an abelian scheme may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties and K3 surfaces.
Projections.
Let formula_106 be a linear subspace; i.e., formula_107 for some linearly independent linear functionals "si". Then the projection from "E" is the (well-defined) morphism
The geometric description of this map is as follows:
Projections can be used to cut down the dimension in which a projective variety is embedded, up to finite morphisms. Start with some projective variety formula_118 If formula_119 the projection from a point not on "X" gives formula_120 Moreover, formula_121 is a finite map to its image. Thus, iterating the procedure, one sees there is a finite map
This result is the projective analog of Noether's normalization lemma. (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 formula_123 In particular, if "X" is normal, then it is the normalization of "H".
Duality and linear system.
While a projective "n"-space formula_1 parameterizes the lines in an affine "n"-space, the dual of it parametrizes the hyperplanes on the projective space, as follows. Fix a field "k". By formula_125, we mean a projective "n"-space
equipped with the construction:
where formula_129 is an "L"-point of formula_125 for a field extension "L" of "k" and formula_131
For each "L", the construction is a bijection between the set of "L"-points of formula_125 and the set of hyperplanes on formula_128. Because of this, the dual projective space formula_125 is said to be the moduli space of hyperplanes on formula_40.
A line in formula_125 is called a pencil: it is a family of hyperplanes on formula_40 parametrized by formula_138.
If "V" is a finite-dimensional vector space over "k", then, for the same reason as above, formula_139 is the space of hyperplanes on formula_140. 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 formula_141 a vector subspace of finite positive dimension. Then there is a map:
determined by the linear system "V", where "B", called the base locus, is the intersection 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 formula_143 on "X" satisfies the following important theorems due to Serre:
These results are proven reducing to the case formula_151 using the isomorphism
where in the right-hand side formula_146 is viewed as a sheaf on the projective space by extension by zero. The result then follows by a direct computation for formula_154 "n" any integer, and for arbitrary formula_143 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 formula_156 is coherent. The same result holds for proper morphisms "f", as can be shown with the aid of Chow's lemma.
Sheaf cohomology 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 of formula_146,
is a well-defined integer (for "X" projective). One can then show formula_159 for some polynomial "P" over rational numbers. Applying this procedure to the structure sheaf formula_160, 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
which is manifestly intrinsic; i.e., independent of the embedding.
The arithmetic genus of a hypersurface of degree "d" is formula_162 in formula_1. In particular, a smooth curve of degree "d" in formula_95 has arithmetic genus formula_165. 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 ω"X", defined as the sheaf of Kähler differentials of top degree (i.e., algebraic "n"-forms), is a line bundle.
Serre duality.
Serre duality states that for any locally free sheaf formula_146 on "X",
where the superscript prime refers to the dual space and formula_168 is the dual sheaf of formula_146.
A generalization to projective, but not necessarily smooth schemes is known as Verdier duality.
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 formula_170. By definition, the geometric genus 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. Since "X" is smooth, there is an isomorphism of groups
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 formula_172. Then the Riemann–Roch theorem states: if "g" is a genus of "X",
for any divisor "D" on "X". By the Serre duality, this is the same as:
which can be readily proved. A generalization of the Riemann-Roch theorem to higher dimension is the Hirzebruch-Riemann-Roch theorem, as well as the far-reaching Grothendieck-Riemann-Roch theorem.
Hilbert schemes.
"Hilbert schemes" 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, i.e., a geometric object whose points parametrize other geometric objects. More precisely, the Hilbert scheme parametrizes closed subvarieties whose Hilbert polynomial equals a prescribed polynomial "P". It is a deep theorem of Grothendieck that there is a scheme formula_175 over "k" such that, for any "k"-scheme "T", there is a bijection
The closed subscheme of formula_177 that corresponds to the identity map formula_178 is called the "universal family".
For formula_179, the Hilbert scheme formula_180 is called the Grassmannian of "r"-planes in formula_1 and, if "X" is a projective scheme, formula_175 is called the Fano scheme of "r"-planes on "X".
Complex projective varieties.
In this section, all algebraic varieties are complex 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, denoted formula_183. Moreover, geometric properties of "X" are reflected by the ones of formula_183. For example, the latter is a complex manifold if and only if "X" is smooth; it is compact if and only if "X" is proper over formula_185.
Relation to complex Kähler manifolds.
Complex projective space is a Kähler manifold. This implies that, for any projective algebraic variety "X", formula_183 is a compact Kähler manifold. The converse is not in general true, but the Kodaira embedding theorem gives a criterion for a Kähler manifold to be projective.
In low dimensions, there are the following results:
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 satisfying certain growth condition is necessarily algebraic: "projective" provides this growth condition. One can deduce from the theorem the following:
Chow's theorem can be shown via Serre's GAGA principle. Its main theorem states:
Complex tori vs. complex abelian varieties.
The complex manifold associated to an abelian variety "A" over formula_185 is a compact complex Lie group. These can be shown to be of the form
and are also referred to as complex tori. Here, "g" is the dimension of the torus and "L" is a lattice (also referred to as period lattice).
According to the uniformization theorem already mentioned above, any torus of dimension 1 arises from an abelian variety of dimension 1, i.e., from an elliptic curve. In fact, the Weierstrass's elliptic function formula_192 attached to "L" satisfies a certain differential equation and as a consequence it defines a closed immersion:
There is a "p"-adic analog, the p-adic uniformization 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 states that for an ample line bundle formula_63 on a smooth projective variety "X" over a field of characteristic zero,
for "i" > 0, or, equivalently by Serre duality formula_196 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.