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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 Unital (geometry) ： ウィキペディア英語版 Unital (geometry) In geometry, a unital is a set of ''n''3 + 1 points arranged into subsets of size ''n'' + 1 so that every pair of distinct points of the set are contained in exactly one subset. ''n'' ≥ 3 is required by some authors to avoid small exceptional cases.〔In particular, 〕 This is equivalent to saying that a unital is a 2-(''n''3 + 1, ''n'' + 1, 1) block design. Some unitals may be embedded in a projective plane of order ''n''2 (the subsets of the design become sets of collinear points in the projective plane). In this case of ''embedded unitals'', every line of the plane intersects the unital in either 1 or ''n'' + 1 points. In the Desarguesian planes, PG(2,''q''2), the classical examples of unitals are given by nondegenerate Hermitian curves. There are also many non-classical examples. The first and the only known unital with non prime power parameters, ''n''=''6'', was constructed by Bhaskar Bagchi and Sunanda Bagchi. It is still unknown if this unital can be embedded in a projective plane of order ''36'', if such a plane exists. ==Classical unitals== We review some terminology used in projective geometry. A correlation of a projective geometry is a bijection on its subspaces that reverses containment. In particular, a correlation interchanges points and hyperplanes. A correlation of order two is called a polarity. A polarity is called a unitary polarity if its associated sesquilinear form ''s'' with companion automorphism ''α'' satisfies: :: ''s''(''u'',''v'') = ''s''(''v'',''u'')''α'' for all vectors ''u'', ''v'' of the underlying vector space. A point is called an absolute point of a polarity if it lies on the image of itself under the polarity. The absolute points of a unitary polarity of the projective geometry PG(''d'',''F''), for some ''d'' ≥ 2, is a nondegenerate Hermitian variety, and if ''d'' = 2 this variety is called a nondegenerate Hermitian curve. In PG(2,''q''2) for some prime power ''q'', the set of points of a nondegenerate Hermitian curve form a unital, which is called a ''classical unital''. Let $\backslash mathcal\; =\; \backslash mathcal(2,q^2)$ be a nondegenerate Hermitian curve in $PG(2,q^2)$ for some prime power $q$. As all nondgenerate Hermitian curves in the same plane are projectively equivalent, $\backslash mathcal$ can be described in terms of homogeneous coordinates as follows: ::$\backslash mathcal\; =\; \backslash \; +\; x\_2^\; =\; 0\; \backslash \}.$ 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「Unital (geometry)」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.