Moduli spaces played a key role in my PhD and I still can't define a moduli space
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First rough definition: a moduli space is the set of isomorphism classes of objects in some category, given the structure of a "space" (e.g. manifold, variety, scheme) in a useful way. (1/n)
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Intuitively I like to think of "The moduli space of X" as "geometric object M where each point of M corresponds to a geometric object of type X". Ex: a circle with radius r is defined by x^2+y^2 = r^2. Thus can think of MC = {r \in RR: r > 0} as "The moduli space of circles".
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Geometric properties of M, (the moduli space of X) tells you a lot about objects of type X. A curve on M corresponds to a deformation of one object of type X to another. E.g. a line segment [a, b] in MC above represents a deformation of a circle of radius a to one of radius b.
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I had a go at this a while back.https://twitter.com/stubborncurias/status/1142546651808194560?s=19 …
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C category of spaces, F:C \to Set a functor. M is a (fine) moduli space if F is representable by M \in C, i.e. if F \iso Hom_C(--, M). Ex.: C =alg. vars. F(S) = family of curves over S of genus g= X \to S smooth of rel dim 1, c_1(X/S)|fibre = 2g-2. M_g = mod. space of curves
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Isn't the technical answer to this that any geometric object "is" a moduli space because it represents its own functor of points?
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