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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 Theorem of Bertini ： ウィキペディア英語版 Theorem of Bertini In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying field, while the extensions require characteristic 0.〔Hartshorne, Ch. III.10.〕 == Statement for hyperplane sections of smooth varieties== Let ''X'' be a smooth quasi-projective variety over an algebraically closed field, embedded in a projective space $\backslash mathbf\; P^n$. Let $|H|$ denote the complete system of hyperplane divisors in $\backslash mathbf\; P^n$. Recall that it is the dual space $(\backslash mathbf\; P^n)^$ of $\backslash mathbf\; P^n$ and is isomorphic to $\backslash mathbf\; P^n$. The theorem of Bertini states that the set of hyperplanes not containing ''X'' and with smooth intersection with ''X'' contains an open dense subset of the total system of divisors $|H|$. The set itself is open if ''X'' is projective. If dim(''X'') ≥ 2, then these intersections (called hyperplane sections of ''X'') are connected, hence irreducible. The theorem hence asserts that a ''general'' hyperplane section not equal to ''X'' is smooth, that is: the property of smoothness is generic. Over an arbitrary field ''k'', there is a dense open subset of the dual space $(\backslash mathbf\; P^n)^$ whose rational points define hyperplanes smooth hyperplane sections of ''X''. When ''k'' is infinite, this open subset then has infinitely many rational points and there are infinitely many smooth hyperplane sections in ''X''. Over a finite field, the above open subset may not contain rational points and in general there is no hyperplanes with smooth intersection with ''X''. However, if we take hypersurfaces of sufficientely big degrees, then the theorem of Bertini holds.〔Bjorn Poonen: (''Bertini Theorems over finite fields'' ), Ann. of Math. 160 (2004).〕 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「Theorem of Bertini」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.