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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 intersection number ： ウィキペディア英語版 intersection number In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem. The intersection number is obvious in certain cases, such as the intersection of ''x''- and ''y''-axes which should be one. The complexity enters when calculating intersections at points of tangency and intersections along positive dimensional sets. For example if a plane is tangent to a surface along a line, the intersection number along the line should be at least two. These questions are discussed systematically in intersection theory. == Definition for Riemann surfaces == Let ''X'' be a Riemann surface. Then the intersection number of two closed curves on ''X'' has a simple definition in terms of an integral. For every closed curve ''c'' on ''X'' (i.e., smooth function $c\; :\; S^1\; \backslash to\; X$), we can associate a differential form $\backslash eta\_c$ with the pleasant property that integrals along ''c'' can be calculated by integrals over ''X'': :$\backslash int\_c\; \backslash alpha\; =\; -\backslash int\; \backslash int\_X\; \backslash alpha\; \backslash wedge\; \backslash eta\_c\; =\; (\backslash alpha,$ *\eta_c), for every closed (1-)differential $\backslash alpha$ on ''X'', where $\backslash wedge$ is the wedge product of differentials, and $$ * is the hodge star. Then the intersection number of two closed curves, ''a'' and ''b'', on ''X'' is defined as :$a\; \backslash cdot\; b\; :=\; \backslash int\; \backslash int\_X\; \backslash eta\_a\; \backslash wedge\; \backslash eta\_b\; =\; (\backslash eta\_a,\; -$ *\eta_b) = -\int_b \eta_a. The $\backslash eta\_c$ have an intuitive definition as follows. They are a sort of dirac delta along the curve ''c'', accomplished by taking the differential of a unit step function that drops from 1 to 0 across ''c''. More formally, we begin by defining for a simple closed curve ''c'' on ''X'', a function ''fc'' by letting $\backslash Omega$ be a small strip around ''c'' in the shape of an annulus. Name the left and right parts of $\backslash Omega\; \backslash setminus\; c$ as $\backslash Omega^$ and $\backslash Omega^$. Then take a smaller sub-strip around ''c'', $\backslash Omega\_0$, with left and right parts $\backslash Omega\_0^$ and $\backslash Omega\_0^$. Then define ''fc'' by :. The definition is then expanded to arbitrary closed curves. Every closed curve ''c'' on ''X'' is homologous to $\backslash sum\_^N\; k\_i\; c\_i$ for some simple closed curves ''ci'', that is, :$\backslash int\_c\; \backslash omega\; =\; \backslash int\_\; \backslash omega\; =\; \backslash sum\_^N\; k\_i\; \backslash int\_\; \backslash omega$, for every differential $\backslash omega$. Define the $\backslash eta\_c$ by :$\backslash eta\_c\; =\; \backslash sum\_^N\; k\_i\; \backslash eta\_$. 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「intersection number」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.