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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク blowing up ： ウィキペディア英語版 blowing up 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. 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 every birational map 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. 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. 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. A blowup can also be called ''monoidal transformation'', ''locally quadratic transformation'', ''dilatation'', σ-''process'', or ''Hopf map''. == The blowup of a point in a plane == 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. 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 P2 at the point ''P'', which we will denote ''X'', is :$X\; =\; \backslash \; \backslash subseteq\; \backslash mathbf^2\; \backslash times\; \backslash mathbf(1,2).$ ''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 P2 that takes the pair $(Q,\; \backslash ell)$ to ''Q''. This morphism is an isomorphism on the open subset of all points $(Q,\; \backslash ell)$ with ''Q'' ≠ ''P'' because the line $\backslash ell$ is determined by those two points. When ''Q'' = ''P'', however, the line $\backslash ell$ can be any line through ''P''. These lines correspond to the space of directions through ''P'', which is isomorphic to P1. This P1 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''. To give coordinates on the blowup, we can write down equations for the above incidence correspondence. Give P2 homogeneous coordinates () in which ''P'' is the point (). By projective duality, G(1,2) is isomorphic to P2, so we may give it homogeneous coordinates (). A line $\backslash ell\_0\; =\; ()$ is the set of all () such that ''X''0''L''0 + ''X''1''L''1 + ''X''2''L''2 = 0. Therefore, the blowup can be described as :$X\; =\; \backslash \; \backslash subseteq\; \backslash mathbf^2\; \backslash times\; \backslash mathbf^2.$ 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'' = (). Write ''x'' and ''y'' for the coordinates on the affine plane ''X''2≠0. The condition ''P'' ∈ $\backslash ell$ implies that ''L''2 = 0, so we may replace the Grassmannian with a P1. Then the blowup is the variety :$\backslash \; \backslash subseteq\; \backslash mathbf^2\; \backslash times\; \backslash mathbf^1.$ It is more common to change coordinates so as to reverse one of the signs. Then the blowup can be written as :$\backslash left\; \backslash \; =\; 0\; \backslash right\; \backslash \}.$ This equation is easier to generalize than the previous one. The blowup can also be described by directly using coordinates on the normal space to the point. Again we work on the affine plane A2. The normal space to the origin is the vector space ''m''/''m''2, 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, :$X\; =\; \backslash operatorname\backslash ,\backslash bigoplus\_^\backslash infty\; \backslash operatorname^r\_\; \backslash mathfrak/\backslash mathfrak^2.$ In this example, this has a concrete description as :$X\; =\; \backslash operatorname\backslash \; k()()/(xz\; -\; yw),$ where ''x'' and ''y'' have degree 0 and ''z'' and ''w'' have degree 1. Over the real or complex numbers, the blowup has a topological description as the connected sum $\backslash mathbf^2\backslash \#\backslash mathbf^2$. Assume that ''P'' is the origin in A2 ⊆ P2, and write ''L'' for the line at infinity. A2 \ has an inversion map ''t'' which sends (''x'', ''y'') to (''x''/(|''x''|2 + |''y''|2), ''y''/(|''x''|2 + |''y''|2)). ''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 P2 → A2 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 P2 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. For CP2 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 $\backslash mathbf^2\backslash \#\backslash overline$, where $\backslash overline$ is CP2 with the opposite of the standard orientation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「blowing up」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.