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In algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. ==Motivation== Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they are isomorphic (that is, geometrically the same). Moduli spaces can be thought of as giving a universal space of parameters for the problem. For example, consider the problem of finding all circles in the Euclidean plane up to congruence. Any circle can be described uniquely by giving three points, but many different sets of three points give the same circle: the correspondence is many-to-one. However, circles are uniquely parameterized by giving their center and radius: this is two real parameters and one positive real parameter. Since we are only interested in circles "up to congruence", we identify circles having different centers but the same radius, and so the radius alone suffices to parameterize the set of interest. The moduli space is therefore the positive real numbers. Moduli spaces often carry natural geometric and topological structures as well. In the example of circles, for instance, the moduli space is not just an abstract set, but the absolute value of the difference of the radii defines a metric for determining when two circles are "close". The geometric structure of moduli spaces locally tells us when two solutions of a geometric classification problem are "close", but generally moduli spaces also have a complicated global structure as well. For example, consider how to describe the collection of lines in R2 which intersect the origin. We want to assign a quantity, a modulus, to each line ''L'' of this family that can uniquely identify it, for example a positive angle θ(''L'') with 0 ≤ θ < π radians, which will yield all lines in R2 which intersect the origin. The set of lines ''L'' just constructed is known as P1(R) and is called the real projective line. We can also describe the collection of lines in R2 that intersect the origin by means of a topological construction. That is, consider S1 ⊂ R2 and notice that to every point ''s'' ∈ S1 that we can identify a line ''L''(''s'') in the collection if the line intersects the origin and ''s''. Yet, this map is two-to-one, so we want to identify ''s'' ~ −''s'' to yield P1(R) ≅ S1/~ where the topology on this space is the quotient topology induced by the quotient map S1 → P1(R). Thus, when we consider P1(R) as a moduli space of lines that intersect the origin in R2, we capture the ways in which the members of the family (lines in the case) can modulate by continuously varying 0 ≤ θ < π. 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「Moduli space」の詳細全文を読む スポンサード リンク
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