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| In [[algebraic geometry]], a '''very ample [[line bundle]]''' is one with enough [[global section]]s to set up an [[embedding]] of its base [[algebraic variety|variety]] or [[manifold (mathematics)|manifold]] <math>M</math> into [[projective space]]. An '''ample line bundle''' is one such that some positive power is very ample. '''Globally generated sheaves''' are those with enough sections to define a [[morphism]] to projective space.
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| ==Introduction==
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| ===Inverse image of line bundle and hyperplane divisors===
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| Given a morphism <math>f\colon X \to Y</math>, any [[vector bundle]] <math>\mathcal F</math> on ''Y'', or more generally any sheaf in <math>\mathcal O_Y</math> modules, ''e.g.'' a [[coherent sheaf]], can be [[pullback|pulled back]] to ''X'', (see [[Inverse image functor]]). This construction preserves the condition of being a [[line bundle]], and more generally the rank.
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| The notions described in this article are related to this construction in the case of morphisms to projective spaces
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| :<math>f\colon X \to \mathbb P^N, </math> and <math>\mathcal F = \mathcal O(1) \in \mathrm{Pic}(\mathbb P^N)</math>, | |
| the line bundle corresponding to the [[hyperplane divisor]], whose [[section (fiber bundle)|section]]s are the 1-homogeneous regular functions. See [[Algebraic geometry of projective spaces#Divisors and twisting sheaves]].
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| === Sheaves generated by their global sections ===
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| Let ''X'' be a [[scheme (mathematics)|scheme]] or a [[complex manifold]] and ''F'' a sheaf on ''X''. One says that ''F'' is '''generated by (finitely many) global sections''' <math> a_i \in F(X)</math>, if every [[stalks of a sheaf|stalk]] of ''F'' is generated as a [[module]] over the stalk of the [[structure sheaf]] by the [[germ (mathematics)|germ]]s of the ''a<sub>i</sub>''. For example, if ''F'' happens to be a line bundle, i.e. locally free of rank 1, this amounts to having finitely many [[global section]]s, such that for any point ''x'' in ''X'', there is at least one section not vanishing at this point. In this case a choice of such global generators ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub> gives a morphism
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| :<math>f\colon X \rightarrow \mathbb{P}^{n},\ x \mapsto [a_0(x): \dotsb : a_n(x)],</math>
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| such that the pullback ''f''*(''O''(1)) is ''F'' (Note that this evaluation makes sense when ''F'' is a subsheaf of the [[constant sheaf]] of [[rational function]]s on ''X''). The converse statement is also true: given such a morphism ''f'', the pullback of ''O''(1) is generated by its global sections (on ''X'').
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| In more generally, a '''sheaf generated by global sections''' is a [[sheaf (mathematics)|sheaf]] ''F'' on a [[locally ringed space]] ''X'', with structure sheaf ''O''<sub>''X''</sub> that is of a rather simple type. Assume ''F'' is a sheaf of [[abelian group]]s. Then it is asserted that if ''A'' is the abelian group of [[global section]]s, i.e.
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| :<math>A = \Gamma(F,X)</math> | |
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| then for any [[open set]] ''U'' of ''X'', ρ(''A'') spans ''F''(''U'') as an ''O''<sub>''U''</sub>-module. Here
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| :<math>\rho = \rho_{X,U}</math>
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| is the restriction map. In words, all sections of ''F'' are locally generated by the global sections.
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| An example of such a sheaf is that associated in [[algebraic geometry]] to an ''R''-module ''M'', ''R'' being any [[commutative ring]], on the [[spectrum of a ring]] ''Spec''(''R'').
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| Another example: according to [[Cartan's theorem A]], any [[coherent sheaf]] on a [[Stein manifold]] is spanned by global sections.
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| === Very ample line bundles ===
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| Given a [[scheme (mathematics)|scheme]] ''X'' over a base scheme ''S'' or a complex manifold, a line bundle (or in other words an [[invertible sheaf]], that is, a locally free sheaf of rank one) ''L'' on ''X'' is said to be '''very ample''', if there is an embedding ''i : X → '''''P'''<sup>''n''</sup><sub>''S''</sub>, the ''n''-dimensional projective space over ''S'' for some ''n'', such that the [[inverse image functor|pullback]] of the [[Serre twist sheaf|standard twisting sheaf]] ''O''(1) on '''P'''<sup>''n''</sup><sub>''S''</sub> is isomorphic to ''L'':
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| :<math> i^{*}(\mathcal{O}(1)) \cong L.</math>
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| Hence this notion is a special case of the previous one, namely a line bundle is very ample if it is globally generated and the morphism given by some global generators is an embedding.
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| Given a very ample sheaf ''L'' on ''X'' and a [[coherent sheaf]] ''F'', a theorem of [[Jean-Pierre Serre|Serre]] shows that (the coherent sheaf) ''F ⊗ L<sup>⊗n</sup>'' is generated by finitely many global sections for sufficiently large ''n''. This in turn implies that global sections and higher (Zariski) [[Sheaf cohomology|cohomology]] groups
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| :<math>H^i(X, F)</math>
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| are finitely generated. This is a distinctive feature of the projective situation. For example, for the affine ''n''-space ''A<sup>n</sup><sub>k</sub>'' over a field ''k'', global sections of the [[structure sheaf]] ''O'' are polynomials in ''n'' variables, thus not a finitely generated ''k''-vector space, whereas for '''P'''<sup>''n''</sup><sub>''k''</sub>, global sections are just constant functions, a one-dimensional ''k''-vector space.
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| == Definitions ==
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| The notion of '''ample line bundles''' ''L'' is slightly weaker than very ample line bundles: a line bundle ''L'' is ample if for any coherent sheaf ''F'' on ''X'', there exists an integer ''n(F)'', such that ''F'' ⊗ ''L''<sup>⊗''n''</sup> is generated by its global sections. | |
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| An equivalent, maybe more intuitive, definition of the ampleness of the line bundle <math>\mathcal L</math> is its having a positive tensorial power that is very ample. In other words, for <math>n \gg 0 </math> there exists a [[projective embedding]] <math>j: X \to \mathbb P^N</math> such that <math>\mathcal L^{\otimes n} = j^* (\mathcal O(1))</math>, that is the zero divisors of global sections of <math>\mathcal L^{\otimes n}</math>
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| are hyperplane sections.
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| This definition makes sense for the underlying ''divisors'' ([[Cartier divisor]]s) <math>D</math>; an ample <math>D</math> is one where <math>nD</math> ''moves in a large enough [[linear system of divisors|linear system]]''. Such divisors form a [[cone (topology)|cone]] in all divisors of those that are, in some sense, ''positive enough''. The relationship with projective space is that the <math>D</math> for a very ample <math>L</math> corresponds to the [[hyperplane section]]s (intersection with some [[hyperplane]]) of the embedded <math>M</math>.
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| The equivalence between the two definitions is credited to [[Jean-Pierre Serre]] in [[Faisceaux algébriques cohérents]].
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| ==Criteria for ampleness of line bundles==
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| ===Intersection theory===
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| {{see|intersection theory#Intersection theory in algebraic geometry}}
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| To decide in practice when a [[Cartier divisor]] ''D'' corresponds to an ample line bundle, there are some geometric criteria.
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| For curves, a divisor ''D'' is very ample if and only if
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| ''l''(''D'') = 2 + ''l''(''D'' − ''A'' − ''B'') whenever ''A'' and ''B'' are points. By the [[Riemann–Roch theorem]] every divisor of degree
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| at least 2''g'' + 1 satisfies this condition so is very ample. This implies that a divisor is ample if and only if it has positive degree. The [[canonical divisor]] of degree 2''g'' − 2 is very ample if and only if the curve is not
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| a [[hyperelliptic curve]].
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| The '''Nakai–Moishezon criterion''' ({{harvnb|Nakai|1963}}, {{harvnb|Moishezon|1964}}) states that a Cartier divisor ''D'' on a proper scheme ''X'' over an [[algebraically closed field]] is ample if and only if ''D''<sup>dim(''Y'')</sup>.''Y'' > 0 for every closed integral [[subscheme]] ''Y'' of ''X''. In the special case of curves this says that a divisor is ample if and only if it has positive degree, and for a smooth projective [[algebraic surface]] ''S'', the Nakai–Moishezon criterion states that ''D'' is ample if and only if its [[self-intersection number]] ''D''.''D'' is strictly positive, and for any [[Irreducible component|irreducible]] curve ''C'' on ''S'' we have ''D''.''C'' > 0.
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| The '''Kleiman condition''' states that for any [[projective variety|projective]] scheme ''X'', a divisor ''D'' on ''X'' is ample if and only if ''D''.''C'' > 0 for any nonzero element ''C'' in the [[closure (topology)|closure]] of NE(''X''), the [[cone of curves]] of ''X''. In other words a divisor is ample if and only if it is in the interior of the real cone generated by [[nef divisor]]s.
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| {{harvtxt|Nagata|1959}} constructed divisors on surfaces that have positive intersection with every curve, but are not ample.
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| This shows that the condition ''D''.''D'' > 0 cannot be omitted in the Nakai–Moishezon criterion, and it is necessary to use the closure of NE(''X'') rather than NE(''X'') in the Kleiman condition.
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| {{harvtxt|Seshadri|1972|loc=Remark 7.1, p. 549}} showed that a line bundle ''L'' on a complete algebraic scheme is ample if and only if there is some positive ε such that
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| deg(''L''|<sub>''C''</sub>) ≥ ε''m''(''C'') for all integral curves ''C'' in ''X'', where ''m''(''C'') is the
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| maximum of the multiplicities at the points of ''C''.
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| ===Sheaf cohomology===
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| The theorem of [[Henri Cartan|Cartan]]-[[Jean-Pierre Serre|Serre]]-[[Grothendieck]] states that for a line bundle <math>\mathcal L</math> on a variety <math>X</math>, the following conditions are equivalent:
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| * <math>\mathcal L</math> is ample
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| * for ''m'' big enough, <math>\mathcal L^{\otimes m}</math> is very ample
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| * for any coherent sheaf <math>\mathcal F</math> on ''X'', the sheaf <math>\mathcal F \otimes \mathcal L^{\otimes m}</math> is generated by global sections, for ''m'' big enough
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| If <math>X</math> is proper over some noetherian ring, this is also equivalent to:
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| * for any coherent sheaf <math>\mathcal F</math> on ''X'', the [[sheaf cohomology|higher cohomology groups]] <math>H^i(X, \mathcal F \otimes \mathcal L^{\otimes m}), \ i \geq 1</math> vanish for ''m'' big enough.
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| ==Generalizations==
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| === Vector bundles of higher rank ===
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| A [[locally free sheaf]] ([[vector bundle]]) <math>F</math> on a variety is called '''ample''' if the [[invertible sheaf]] <math>\mathcal{O}(1)</math> on <math>\mathbb{P}(F)</math> is ample {{harvtxt|Hartshorne|1966}}.
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| Ample vector bundles inherit many of the properties of ample line bundles.
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| ===Big line bundles===
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| {{main| Iitaka dimension}}
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| An important generalization, notably in [[birational geometry]], is that of a '''big line bundle'''. A line bundle <math>\mathcal L</math> on ''X'' is said to be big if the equivalent following conditions are satisfied:
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| *<math>\mathcal L</math> is the tensor product of an ample line bundle and an effective line bundle
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| *the [[Hilbert polynomial]] of the finitely generated [[graded ring]] <math>\bigoplus_{k=0}^\infty \Gamma (X, \mathcal L ^{\otimes k})</math> has degree the dimension of ''X''
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| *the rational mapping of the [[linear system of divisors|total system of divisors]] <math>X \to \mathbb P \Gamma (X, \mathcal L^{\otimes k})</math> is [[birational]] on its image for <math>k \gg 0</math>.
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| The interest of this notion is its stability with respect to rational transformations.
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| ==See also==
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| ===General algebraic geometry===
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| *[[Cartier divisor]]
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| *[[Algebraic geometry of projective spaces]]
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| *[[Fano variety]]: a variety whose [[canonical bundle|canonical line bundle]] is anti-ample
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| ===Ampleness in complex geometry===
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| *[[Holomorphic vector bundle]]
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| *A line bundle is ample if and only if its [[Chern class]] is a Kahler class.
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| *[[Kodaira embedding theorem]]: for compact complex manifolds, ampleness and positivity coincide.
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| *[[Lefschetz hyperplane theorem]]: the study of very ample line bundles on complex projective manifolds gives strong topological information
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| ==References==
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| ===Study references===
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| * {{Citation | last1=Hartshorne | first1=Robin | author1-link= Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | year=1977}}
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| * {{Citation | last1=Lazarsfeld | first1=Robert | author1-link= Robert Lazarsfeld | title=[[Positivity in Algebraic Geometry (book)|Positivity in Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin | year=2004}}
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| * The slides on ampleness in Vladimir Lazić's [http://www2.imperial.ac.uk/~vlazic/AGlect11.pdf Lectures on algebraic geometry]
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| ===Research texts===
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| *{{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Ample vector bundles | url=http://www.numdam.org/item?id=PMIHES_1966__29__63_0 | mr=0193092 | year=1966 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=29 | pages=63–94}}
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| *{{Citation | doi=10.2307/1970447 | last1=Kleiman | first1=Steven L. | author1-link=Steven Kleiman | title=Toward a numerical theory of ampleness | jstor=1970447 | mr=0206009 | year=1966 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=84 | pages=293–344 | issue=3 | publisher=Annals of Mathematics}}
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| *{{Citation | last1=Moishezon | first1=B. G. | authorlink1 = Boris Moishezon | title=A projectivity criterion of complete algebraic abstract varieties | mr=0160782 | year=1964 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=28 | pages=179–224}}
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| * {{Citation | last1=Nagata | first1=Masayoshi | author1-link= Masayoshi Nagata | title=On the 14th problem of Hilbert | mr=0154867 | year=1959 | journal=[[American Journal of Mathematics]] | volume=81 | pages=766–772 | doi=10.2307/2372927 | jstor=2372927 | issue=3 | publisher=The Johns Hopkins University Press}}
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| *{{Citation | doi=10.2307/2373180 | last1=Nakai | first1=Yoshikazu | title=A criterion of an ample sheaf on a projective scheme | jstor=2373180 | mr=0151461 | year=1963 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=85 | pages=14–26 | issue=1 | publisher=The Johns Hopkins University Press}}
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| *{{Citation | doi=10.2307/1970870 | last1=Seshadri | first1=C. S. | title=Quotient spaces modulo reductive algebraic groups | jstor=1970870 | mr=0309940 | year=1972 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=95 | pages=511–556 | issue=3 | publisher=Annals of Mathematics}}
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| [[Category:Vector bundles]]
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| [[Category:Algebraic geometry]]
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| [[Category:Geometry of divisors]]
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