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In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point ''P'' on an algebraic variety ''V'' (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations. == Motivation == For example, suppose given a plane curve ''C'' defined by a polynomial equation :''F(X,Y) = 0'' and take ''P'' to be the origin (0,0). Erasing terms of higher order than 1 would produce a 'linearised' equation reading :''L(X,Y) = 0'' in which all terms ''XaYb'' have been discarded if ''a + b > 1''. We have two cases: ''L'' may be 0, or it may be the equation of a line. In the first case the (Zariski) tangent space to ''C'' at (0,0) is the whole plane, considered as a two-dimensional affine space. In the second case, the tangent space is that line, considered as affine space. (The question of the origin comes up, when we take ''P'' as a general point on ''C''; it is better to say 'affine space' and then note that ''P'' is a natural origin, rather than insist directly that it is a vector space.) It is easy to see that over the real field we can obtain ''L'' in terms of the first partial derivatives of ''F''. When those both are 0 at ''P'', we have a singular point (double point, cusp or something more complicated). The general definition is that ''singular points'' of ''C'' are the cases when the tangent space has dimension 2. 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「Zariski tangent space」の詳細全文を読む スポンサード リンク
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