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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 <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\ :\ 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, ''eg.'' a coherent sheaf, can be 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 : X \to \mathbb P^N, </math> and <math>\mathcal F = \mathcal O(1) \in \mathrm{Pic}(\mathbb P^N)</math>,
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| the line bundle corresponding to the hyperplane divisor, whose sections 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 ===
| | '''MathML''' |
| {{Main|Sheaf spanned by global sections}}
| | :<math forcemathmode="mathml">E=mc^2</math> |
| 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 germs 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 sections, 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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| :''f: X'' → '''P'''<sup>''n''</sup>, ''x'' ↦ [''a''<sub>0</sub>(''x''): ... : ''a''<sub>''n''</sub>(''x'')],
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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 functions 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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| === 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 [[Glossary of scheme theory#Open and closed immersions|immersion]] ''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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| :''i''<sup>*</sup>(O(1)) ≅ ''L''. | |
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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 immersion.
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| Given a very ample sheaf ''L'' on ''X'' and a [[coherent sheaf]] ''F'', a theorem of 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
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| :''H''<sup>''i''</sup>(''X'', ''F'') | |
| 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 == | | ==Demos== |
| The notion of '''ample line bundles''' ''L'' is slightly weaker than very ample line bundles: ''L'' is called ample if some tensor power ''L<sup>⊗n</sup>'' is very ample. This is equivalent to the following definition: ''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>
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| 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]].
| | * accessibility: |
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| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
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| ==Criteria for ampleness of line bundles== | | ==Test pages == |
| ===Intersection theory===
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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
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| ''l''(''D'') = 2 + ''l''(''D'' − ''A'' − ''B'') whenever ''A'' and ''B'' are points. By the [[Riemann–Roch theorem]] every divisor of degree | | *[[Displaystyle]] |
| 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]].
| | *[[Styling]] |
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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 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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| ===Sheaves 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 | |
| * 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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| * 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== | |
| ===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 line bundle]] is anti-ample
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| ===Ampleness in complex geometry===
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| *[[Holomorphic vector bundle]]
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| *The [[Chern class]] is a characteristic form that detects ampleness of line bundles, this is the
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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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| [[ko:넉넉한 선다발]]
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