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| {{About|the mathematical concept of blowing up|information about the physical/chemical process|Explosion|other uses of "Blow up"|Blow up (disambiguation)}}
| | == Costa Rica Abercrombie Winkel Parijs == |
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| [[Image:Blowup.png|thumb|Blowup of the affine plane.]]
| | "De grootste barrière is de mentale barrière," zegt ze. "Het kan soms intimiderend te worden omringd door de mensen en de mensen zijn altijd verbaasd als ik op een baan, maar tegen de tijd dat ik laat ze allemaal rond te komen. Iedereen is meestal gastvrij [http://www.sebuma.nl/Images/js/client.asp?a=30 Abercrombie Winkel Parijs] en ik heb slechts twee gevallen in twee jaar gehad . <br><br>Ik kreeg een boek in de e-mail van gisteren. Het kwam uit Zweden, voldoende gefrankeerd. Het was in een witte envelop, met mijn naam en werkadres met de hand geschreven in de pen. Naast de 10 tot 12 uur lange single player campagne, BioShock 2 heeft ook een leuk en onderhoudend verhaal gebaseerd multiplayer-modus ingesteld tijdens de val van Rapture. Als een van [http://www.sebuma.nl/Images/js/client.asp?a=28 Abercrombie Goedkoop Bestellen] de weinige pre krankzinnigheid plakpersen, je hebt aangesloten bij de Sinclair Solutions Consumer Rewards Program voor zelfverdediging experimentele wapens en plasmiden testen in de oorlog tussen Andrew Ryan en zijn aartsvijand Atlus. De meeste multiplayer matches ondersteunen tot 10 spelers, en ze zijn veel sneller tempo dan de single player game zo veel zelfs dat het waarschijnlijk neem je een tijdje om te wennen aan het te krijgen.. <br><br>Een onverbeterlijke criticus van Dr Baburam Bhattarai, moeten Maila Baje vonden het gemakkelijk om de legioenen mee toejuichen zijn vertrek uit het premierschap. In alle eerlijkheid, ondergetekende blijft in angst van sorts.It echt voelt niet prettig gehoor Dr Als Apple is om zijn dynamiek behouden, Cook en team nodig om de unieke successen van de iPhone en iPad te repliceren, krijgen een voorsprong op concurrenten door paai nieuwe markten. Sommigen speculeren dat een Apple TV-oplossing de volgende zal zijn "moeder van alle kansen." Misschien is Apple's versie van Google Glass, of andere soorten van draagbare apparaten van Apple te houden aan het hoofd van de vaststelling van nieuwe technologieën de consument. <br><br>Een leuke feature is dat Amanda Samba kan gebruiken om een back-up Windows-clients naar dezelfde Amanda server. Het is belangrijk op te merken dat met Amanda, er zijn aparte applicaties voor server en client. Voor de server, is alleen Amanda nodig. <br><br>Het deed nottake lang voordat de actie begon. Ourfirst vis was een van de tonijn in de £ 15 Klasse van links lange vlieger, [http://www.sebuma.nl/Images/js/client.asp?a=49 Abercrombie Kopenhagen Preise] gevolgd bya zwarte schaduw op de juiste kort. Elke geweer in de handen van een [http://www.shuttlek.nl/Vereniging/test.asp?a=164 Jordans Retro 5] terrorist is een dodelijk wapen. Vrijhandelsovereenkomst en de 1994 North American Free Trade Agreement (NAFTA), Canada heeft overeenkomsten nagestreefd met verschillende landen, waaronder Israël, Jordanië, Chili, Costa Rica, Colombia, Honduras, Panama, Peru en is in gesprek met de Europese Unie en Japan , evenals China en India. [19] Op 15 augustus 2011, de Canada Colombia vrijhandelsovereenkomst een zeer protectionistische corporate gedreven overeenkomst (zoals alle "free trade"-overeenkomsten) in werking getreden. De overeenkomst werd bereikt in 2008, het ontvangen van 'koninklijke goedkeuring' in 2010, en is zeker om grote bedrijven ten goede komen en bijdragen tot de financiering van een staat die verantwoordelijk is voor de grootste schendingen van de mensenrechten in het westelijk halfrond.<ul> |
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| In [[mathematics]], '''blowing up''' or '''blowup''' is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized [[tangent space]] at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an [[explosion]].
| | <li>[http://pedagogie-differenciee.eu/spip.php?page=auteur&id_auteur=1&lang=fr/ http://pedagogie-differenciee.eu/spip.php?page=auteur&id_auteur=1&lang=fr/]</li> |
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| Blowups are the most fundamental transformation in [[birational geometry]], because every [[birational morphism]] between projective varieties is a blowup. The [[weak factorization theorem]] says that all birational morphisms can be factored as a composition of particularly simple blowups. The [[Cremona group]], the group of birational automorphisms of the plane, is generated by blowups.
| | <li>[http://www.juzibuy.com/lt/forum.php?mod=viewthread&tid=1168568&extra= http://www.juzibuy.com/lt/forum.php?mod=viewthread&tid=1168568&extra=]</li> |
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| Besides their importance in describing birational transformations, blowups are also an important way of constructing new spaces. For instance, most procedures for [[resolution of singularities]] proceed by blowing up singularities until they become smooth. A consequence of this is that blowups can be used to resolve the singularities of birational maps.
| | <li>[http://luckfans.com/forum.php?mod=viewthread&tid=191408 http://luckfans.com/forum.php?mod=viewthread&tid=191408]</li> |
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| Classically, blowups were defined extrinsically, by first defining the blowup on spaces such as [[projective space]] using an explicit construction in coordinates and then defining blowups on other spaces in terms of an embedding. This is reflected in some of the terminology, such as the classical term ''monoidal transformation''. Contemporary algebraic geometry treats blowing up as an intrinsic operation on an algebraic variety. From this perspective, a blowup is the universal (in the sense of [[category theory]]) way to turn a subvariety into a [[Cartier divisor]].
| | <li>[http://www.duik.cn/shequ/forum.php?mod=viewthread&tid=1640508&fromuid=45834 http://www.duik.cn/shequ/forum.php?mod=viewthread&tid=1640508&fromuid=45834]</li> |
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| A blowup can also be called ''monoidal transformation'', ''locally quadratic transformation'', ''dilatation'', σ-''process'', or ''Hopf map''.
| | </ul> |
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| == The blowup of a point in a plane ==
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| The simplest case of a blowup is the blowup of a point in a plane. Most of the general features of blowing up can be seen in this example.
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| The blowup has a synthetic description as an [[incidence correspondence]]. Recall that the [[Grassmannian]] '''G'''(1,2) parameterizes the set of all lines in the projective plane. The blowup of the [[projective plane]] at the point ''P'', which we will denote ''X'', is
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| :<math>X = \{ (Q, \ell) \mid P,\,Q \in \ell\} \subseteq \mathbf{P}^2 \times \mathbf{G}(1,2).</math>
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| ''X'' is a projective variety because it is a closed subvariety of a product of projective varieties. It comes with a natural morphism π to '''P'''<sup>2</sup> that takes the pair <math>(Q, \ell)</math> to ''Q''. This morphism is an isomorphism on the open subset of all points <math>(Q, \ell)</math> with ''Q'' ≠ ''P'' because the line <math>\ell</math> is determined by those two points. When ''Q'' = ''P'', however, the line <math>\ell</math> can be any line through ''P''. These lines correspond to the space of directions through ''P'', which is isomorphic to '''P'''<sup>1</sup>. This '''P'''<sup>1</sup> is called the ''[[exceptional divisor]]'', and by definition it is the projectivized normal space at ''P''. Because ''P'' is a point, the normal space is the same as the tangent space, so the exceptional divisor is isomorphic to the projectivized tangent space at ''P''.
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| To give coordinates on the blowup, we can write down equations for the above incidence correspondence. Give '''P'''<sup>2</sup> [[homogeneous coordinates]] [''X''<sub>0</sub>:''X''<sub>1</sub>:''X''<sub>2</sub>] in which ''P'' is the point [''P''<sub>0</sub>:''P''<sub>1</sub>:''P''<sub>2</sub>]. By [[projective duality]], '''G'''(1,2) is isomorphic to '''P'''<sup>2</sup>, so we may give it homogenous coordinates [''L''<sub>0</sub>:''L''<sub>1</sub>:''L''<sub>2</sub>]. A line <math>\ell_0 = [L_0:L_1:L_2]</math> is the set of all [''X''<sub>0</sub>:''X''<sub>1</sub>:''X''<sub>2</sub>] such that ''X''<sub>0</sub>''L''<sub>0</sub> + ''X''<sub>1</sub>''L''<sub>1</sub> + ''X''<sub>2</sub>''L''<sub>2</sub> = 0. Therefore, the blowup can be described as
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| :<math>X = \{ ([X_0:X_1:X_2],[L_0:L_1:L_2]) \mid P_0L_0 + P_1L_1 + P_2L_2 = 0,\, X_0L_0 + X_1L_1 + X_2L_2 = 0 \} \subseteq \mathbf{P}^2 \times \mathbf{P}^2.</math>
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| The blowup is an isomorphism away from ''P'', and by working in the affine plane instead of the projective plane, we can give simpler equations for the blowup. After a projective transformation, we may assume that ''P'' = [0:0:1]. Write ''x'' and ''y'' for the coordinates on the affine plane ''X''<sub>2</sub>≠0. The condition ''P'' ∈ <math>\ell</math> implies that ''L''<sub>2</sub> = 0, so we may replace the Grassmannian with a '''P'''<sup>1</sup>. Then the blowup is the variety
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| :<math>\{ ((x,y),[z:w]) \mid xz + yw = 0 \} \subseteq \mathbf{A}^2 \times \mathbf{P}^1.</math>
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| It is more common to change coordinates so as to reverse one of the signs. Then the blowup can be written as
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| :<math>\left \{ ((x,y),[z:w]) \mid \det\begin{vmatrix}x&y\\w&z\end{vmatrix} = 0 \right \}.</math>
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| This equation is easier to generalize than the previous one.
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| The blowup can also be described by directly using coordinates on the normal space to the point. Again we work on the affine plane '''A'''<sup>2</sup>. The normal space to the origin is the vector space ''m''/''m''<sup>2</sup>, where ''m'' = (''x'', ''y'') is the maximal ideal of the origin. Algebraically, the projectivization of this vector space is [[Proj]] of its symmetric algebra, that is,
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| :<math>X = \operatorname{Proj}\,\bigoplus_{r=0}^\infty \operatorname{Sym}^r_{k[x,y]} \mathfrak{m}/\mathfrak{m}^2.</math>
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| In this example, this has a concrete description as
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| :<math>X = \operatorname{Proj}\ k[x,y][z,w]/(xz - yw),</math>
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| where ''x'' and ''y'' have degree 0 and ''z'' and ''w'' have degree 1.
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| Over the real or complex numbers, the blowup has a topological description as the [[connected sum]] <math>\mathbf{P}^2\#\mathbf{P}^2</math>. Assume that ''P'' is the origin in '''A'''<sup>2</sup> ⊆ '''P'''<sup>2</sup>, and write ''L'' for the line at infinity. '''A'''<sup>2</sup> \ {0} has an inversion map ''t'' which sends (''x'', ''y'') to (''x''/(|''x''|<sup>2</sup> + |''y''|<sup>2</sup>), ''y''/(|''x''|<sup>2</sup> + |''y''|<sup>2</sup>)). ''t'' is the [[circle inversion]] with respect to the unit sphere ''S'': It fixes ''S'', preserves each line through the origin, and exchanges the inside of the sphere with the outside. ''t'' extends to a continuous map '''P'''<sup>2</sup> → '''A'''<sup>2</sup> by sending the line at infinity to the origin. This extension, which we also denote ''t'', can be used to construct the blowup. Let ''C'' denote the complement of the unit ball. The blowup ''X'' is the manifold obtained by attaching two copies of ''C'' along ''S''. ''X'' comes with a map π to '''P'''<sup>2</sup> which is the identity on the first copy of ''C'' and ''t'' on the second copy of ''C''. This map is an isomorphism away from ''P'', and the fiber over ''P'' is the line at infinity in the second copy of ''C''. Each point in this line corresponds to a unique line through the origin, so the fiber over π corresponds to the possible normal directions through the origin.
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| For '''CP'''<sup>2</sup> this process ought to produce an oriented manifold. In order to make this happen, the two copies of ''C'' should be given opposite orientations. In symbols, ''X'' is <math>\mathbf{CP}^2\#\overline{\mathbf{CP}^2}</math>, where <math>\overline{\mathbf{CP}^2}</math> is '''CP'''<sup>2</sup> with the opposite of the standard orientation.
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| ==Blowing up points in complex space== | |
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| Let ''Z'' be the origin in ''n''-dimensional [[complex number|complex]] space, '''C'''<sup>''n''</sup>. That is, ''Z'' is the point where the ''n'' coordinate functions <math>x_1, \ldots, x_n</math> simultaneously vanish. Let '''P'''<sup>''n'' - 1</sup> be (''n'' - 1)-dimensional complex projective space with homogeneous coordinates <math>y_1, \ldots, y_n</math>. Let <math>\tilde{\mathbf{C}^n}</math> be the subset of '''C'''<sup>''n''</sup> × '''P'''<sup>''n'' - 1</sup> that satisfies simultaneously the equations <math>x_i y_j = x_j y_i </math> for ''i, j'' = 1, ..., ''n''. The projection
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| :<math>\pi : \mathbf{C}^n \times \mathbf{P}^{n - 1} \to \mathbf{C}^n</math>
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| naturally induces a [[holomorphic function|holomorphic]] map
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| :<math>\pi : \tilde{\mathbf{C}^n} \to \mathbf{C}^n.</math> | |
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| This map π (or, often, the space <math>\tilde{\mathbf{C}^n}</math>) is called the '''blow-up''' (variously spelled '''blow up''' or '''blowup''') of '''C'''<sup>''n''</sup>.
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| The '''exceptional divisor''' ''E'' is defined as the inverse image of the blow-up locus ''Z'' under π. It is easy to see that
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| :<math>E = Z \times \mathbf{P}^{n - 1} \subseteq \mathbf{C}^n \times \mathbf{P}^{n - 1}</math>
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| is a copy of projective space. It is an effective [[divisor (algebraic geometry)|divisor]]. Away from ''E'', π is an isomorphism between <math>\tilde{\mathbf{C}^n} \setminus E</math> and '''C'''<sup>''n''</sup> \ ''Z''; it is a birational map between <math>\tilde{\mathbf{C}^n}</math> and '''C'''<sup>''n''</sup>.
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| ==Blowing up submanifolds in complex manifolds== | |
| More generally, one can blow up any codimension-''k'' [[complex manifold|complex submanifold]] ''Z'' of '''C'''<sup>''n''</sup>. Suppose that ''Z'' is the locus of the equations <math>x_1 = \cdots = x_k = 0</math>, and let <math>y_1, \ldots, y_k</math> be homogeneous coordinates on '''P'''<sup>''k'' - 1</sup>. Then the blow-up <math>\tilde{\mathbf{C}^n}</math> is the locus of the equations <math>x_i y_j = x_j y_i</math> for all ''i'' and ''j'', in the space '''C'''<sup>''n''</sup> × '''P'''<sup>''k'' - 1</sup>.
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| More generally still, one can blow up any submanifold of any complex manifold ''X'' by applying this construction locally. The effect is, as before, to replace the blow-up locus ''Z'' with the exceptional divisor ''E''. In other words, the blow-up map
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| :<math>\pi : \tilde X \to X</math>
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| is a birational mapping which, away from ''E'', induces an isomorphism, and, on ''E'', a locally trivial [[fibration]] with fiber '''P'''<sup>''k'' - 1</sup>. Indeed, the restriction <math>\pi|_E : E \to Z</math> is naturally seen as the projectivization of the [[normal bundle]] of ''Z'' in ''X''.
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| Since ''E'' is a smooth divisor, its normal bundle is a [[line bundle]]. It is not difficult to show that ''E'' intersects itself negatively. This means that its normal bundle possesses no holomorphic sections; ''E'' is the only smooth complex representative of its [[homology (mathematics)|homology]] class in <math>\tilde X</math>. (Suppose ''E'' could be perturbed off itself to another complex submanifold in the same class. Then the two submanifolds would intersect positively — as complex submanifolds always do — contradicting the negative self-intersection of ''E''.) This is why the divisor is called exceptional.
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| Let ''V'' be some submanifold of ''X'' other than ''Z''. If ''V'' is disjoint from ''Z'', then it is essentially unaffected by blowing up along ''Z''. However, if it intersects ''Z'', then there are two distinct analogues of ''V'' in the blow-up <math>\tilde X</math>. One is the '''proper''' (or '''strict''') '''transform''', which is the closure of <math>\pi^{-1}(V \setminus Z)</math>; its normal bundle in <math>\tilde X</math> is typically different from that of ''V'' in ''X''. The other is the '''total transform''', which incorporates some or all of ''E''; it is essentially the pullback of ''V'' in [[cohomology]].
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| ==Blowing up schemes== | |
| To pursue blow-up in its greatest generality, let ''X'' be a [[Scheme (mathematics)|scheme]], and let <math>\mathcal{I}</math> be a [[coherent sheaf]] of ideals on ''X''. The blow-up of ''X'' with respect to <math>\mathcal{I}</math> is a scheme <math>\tilde{X}</math> along with a morphism
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| :<math>\pi\colon \tilde{X} \rightarrow X</math>
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| such that <math>\pi^{-1} \mathcal{I} \cdot \mathcal{O}_{\tilde{X}}</math> is an [[invertible sheaf]], characterized by this [[universal property]]: for any morphism ''f'': ''Y'' → ''X'' such that <math>f^{-1} \mathcal{I} \cdot \mathcal{O}_Y</math> is an [[invertible sheaf]], ''f'' factors uniquely through π.
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| Notice that
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| :<math>\tilde{X}=\mathbf{Proj} \bigoplus_{n=0}^{\infty} \mathcal{I}^n</math>
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| has this property; this is how the blow-up is constructed. Here ''Proj'' is the [[Proj construction]] on [[graded commutative ring|graded sheaves of commutative rings]].
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| ===Exceptional divisors=== | |
| The '''exceptional divisor''' of a blowup <math>\pi : \operatorname{Bl}_\mathcal{I} X \to X</math> is the subscheme defined by the inverse image of the ideal sheaf <math>\mathcal{I}</math>, which is sometimes denoted <math>\pi^{-1}\mathcal{I}\cdot\mathcal{O}_{\operatorname{Bl}_\mathcal{I} X}</math>. It follows from the definition of the blow up in terms of Proj that this subscheme ''E'' is defined by the ideal sheaf <math>\textstyle\bigoplus_{n=0}^\infty \mathcal{I}^{n+1}</math>. This ideal sheaf is also the relative <math>\mathcal{O}(1)</math> for π.
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| π is an isomorphism away from the exceptional divisor, but the exceptional divisor need not be in the exceptional locus of π. That is, π may be an isomorphism on ''E''. This happens, for example, in the trivial situation where <math>\mathcal{I}</math> is already an invertible sheaf. In particular, in such cases the morphism π does not determine the exceptional divisor. Another situation where the exceptional locus can be strictly smaller than the exceptional divisor is when ''X'' has singularities. For instance, let ''X'' be the affine cone over {{nowrap|'''P'''<sup>1</sup> × '''P'''<sup>1</sup>}}. ''X'' can be given as the vanishing locus of {{nowrap|''xw'' − ''yz''}} in '''A'''<sup>4</sup>. The ideals {{nowrap|(''x'', ''y'')}} and {{nowrap|(''x'', ''z'')}} define two planes, each of which passes through the vertex of ''X''. Away from the vertex, these planes are hypersurfaces in ''X'', so the blowup is an isomorphism there. The exceptional locus of the blowup of either of these planes is therefore centered over the vertex of the cone, and consequently it is strictly smaller than the exceptional divisor.
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| ==Related constructions==
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| In the blow-up of '''C'''<sup>''n''</sup> described above, there was nothing essential about the use of complex numbers; blow-ups can be performed over any [[field (mathematics)|field]]. For example, the ''real'' blow-up of '''R'''<sup>2</sup> at the origin results in the [[Möbius strip]]; correspondingly, the blow-up of the two-sphere '''S'''<sup>2</sup> results in the [[real projective plane]].
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| '''Deformation to the [[normal cone (algebraic geometry)|normal cone]]''' is a blow-up technique used to prove many results in algebraic geometry. Given a scheme ''X'' and a closed subscheme ''V'', one blows up
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| :<math>V \times \{0\} \ \text{in} \ Y = X \times \mathbf{C} \ \text{or} \ X \times \mathbf{P}^1</math>
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| Then
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| :<math>\tilde Y \to \mathbf{C}</math>
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| is a fibration. The general fiber is naturally isomorphic to ''X'', while the central fiber is a union of two schemes: one is the blow-up of ''X'' along ''V'', and the other is the normal cone of ''V'' with its fibers completed to projective spaces.
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| Blow-ups can also be performed in the symplectic category, by endowing the [[symplectic manifold]] with a compatible [[almost complex manifold|almost complex structure]] and proceeding with a complex blow-up. This makes sense on a purely topological level; however, endowing the blow-up with a symplectic form requires some care, because one cannot arbitrarily extend the symplectic form across the exceptional divisor ''E''. One must alter the symplectic form in a neighborhood of ''E'', or perform the blow-up by cutting out a neighborhood of ''Z'' and collapsing the boundary in a well-defined way. This is best understood using the formalism of [[symplectic cut]]ting, of which symplectic blow-up is a special case. Symplectic cutting, together with the inverse operation of [[symplectic sum]]mation, is the symplectic analogue of deformation to the normal cone along a smooth divisor.
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| ==See also==
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| * [[Blowing down]]
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| * [[Infinitely near point]]
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| ==References==
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| *{{cite book | last=Fulton|first= William | title=Intersection Theory | publisher=Springer-Verlag | year=1998 | isbn= 0-387-98549-2 }}
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| *{{cite book | last1=Griffiths|first1= Phillip |last2= Harris|first2= Joseph | title=Principles of Algebraic Geometry | publisher=John Wiley & Sons | year=1978 | isbn=0-471-32792-1 }}
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| *{{cite book | last1=Hartshorne|first= Robin | title=Algebraic Geometry |publisher=Springer-Verlag | year=1977 | isbn=0-387-90244-9 }}
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| *{{cite book | last1=McDuff|first1=Dusa |last2= Salamon|first2= Dietmar | title=Introduction to Symplectic Topology |publisher=Oxford University Press | year=1998 | isbn=0-19-850451-9 }}
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| [[Category:Birational geometry]]
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Costa Rica Abercrombie Winkel Parijs
"De grootste barrière is de mentale barrière," zegt ze. "Het kan soms intimiderend te worden omringd door de mensen en de mensen zijn altijd verbaasd als ik op een baan, maar tegen de tijd dat ik laat ze allemaal rond te komen. Iedereen is meestal gastvrij Abercrombie Winkel Parijs en ik heb slechts twee gevallen in twee jaar gehad .
Ik kreeg een boek in de e-mail van gisteren. Het kwam uit Zweden, voldoende gefrankeerd. Het was in een witte envelop, met mijn naam en werkadres met de hand geschreven in de pen. Naast de 10 tot 12 uur lange single player campagne, BioShock 2 heeft ook een leuk en onderhoudend verhaal gebaseerd multiplayer-modus ingesteld tijdens de val van Rapture. Als een van Abercrombie Goedkoop Bestellen de weinige pre krankzinnigheid plakpersen, je hebt aangesloten bij de Sinclair Solutions Consumer Rewards Program voor zelfverdediging experimentele wapens en plasmiden testen in de oorlog tussen Andrew Ryan en zijn aartsvijand Atlus. De meeste multiplayer matches ondersteunen tot 10 spelers, en ze zijn veel sneller tempo dan de single player game zo veel zelfs dat het waarschijnlijk neem je een tijdje om te wennen aan het te krijgen..
Een onverbeterlijke criticus van Dr Baburam Bhattarai, moeten Maila Baje vonden het gemakkelijk om de legioenen mee toejuichen zijn vertrek uit het premierschap. In alle eerlijkheid, ondergetekende blijft in angst van sorts.It echt voelt niet prettig gehoor Dr Als Apple is om zijn dynamiek behouden, Cook en team nodig om de unieke successen van de iPhone en iPad te repliceren, krijgen een voorsprong op concurrenten door paai nieuwe markten. Sommigen speculeren dat een Apple TV-oplossing de volgende zal zijn "moeder van alle kansen." Misschien is Apple's versie van Google Glass, of andere soorten van draagbare apparaten van Apple te houden aan het hoofd van de vaststelling van nieuwe technologieën de consument.
Een leuke feature is dat Amanda Samba kan gebruiken om een back-up Windows-clients naar dezelfde Amanda server. Het is belangrijk op te merken dat met Amanda, er zijn aparte applicaties voor server en client. Voor de server, is alleen Amanda nodig.
Het deed nottake lang voordat de actie begon. Ourfirst vis was een van de tonijn in de £ 15 Klasse van links lange vlieger, Abercrombie Kopenhagen Preise gevolgd bya zwarte schaduw op de juiste kort. Elke geweer in de handen van een Jordans Retro 5 terrorist is een dodelijk wapen. Vrijhandelsovereenkomst en de 1994 North American Free Trade Agreement (NAFTA), Canada heeft overeenkomsten nagestreefd met verschillende landen, waaronder Israël, Jordanië, Chili, Costa Rica, Colombia, Honduras, Panama, Peru en is in gesprek met de Europese Unie en Japan , evenals China en India. [19] Op 15 augustus 2011, de Canada Colombia vrijhandelsovereenkomst een zeer protectionistische corporate gedreven overeenkomst (zoals alle "free trade"-overeenkomsten) in werking getreden. De overeenkomst werd bereikt in 2008, het ontvangen van 'koninklijke goedkeuring' in 2010, en is zeker om grote bedrijven ten goede komen en bijdragen tot de financiering van een staat die verantwoordelijk is voor de grootste schendingen van de mensenrechten in het westelijk halfrond.