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        Questions tagged [moduli-spaces]

        Given a concrete category C, with objects denoted Obj(C), and an equivalence relation ~ on Obj(C) given by morphisms in C. The moduli set for Obj(C) is the set of equivalence classes with respect to ~; denoted Iso(C). When Iso(C) is an object in the category Top, then the moduli set is called a moduli space.

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        What is the analogy between the moduli of shtukas and Shimura varieties?

        I have heard that moduli spaces of shtukas are supposed to be the analogue of Shimura varieties in the setting of function fields. Could someone more knowledgeable about these objects explain how this ...
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        The moduli scheme of “$\nu$-canonically embeded curves”

        This is related to the proposition 5.1. of Mumford's GIT. It states that: There is a unique subscheme $H$ in the Hilbert scheme $Hilb_{\mathbb{P}^n}^{P(x)}$ such that, for any morphism $f : S \to ...
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        The moduli scheme of smooth curves of given genus is irreducible

        I've heard that Deligne-Mumford's "the irreducibility..." showed this first. But I think that Mumford's "Geometric invariant theory" has its proof. The proof is as follows: Let $H$ be the scheme that ...
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        Level structures in deformation spaces of $p$-divisible groups

        I am reading (parts of) the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein, and I am stuck at understanding the definition of level structures in Rapoport-Zink spaces (cf. Definition ...
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        142 views

        Torsor descriptions of $Bun_G$

        The moduli stack $Bun_{SL_n}X$ of $SL_n$-principal bundles over a projective curve $X$ is a $\mathbb{G}_m$-torsor over the sublocus of $Bun_{GL_n}X$ corresponding to vector bundles with zero ...
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        39 views

        Complexifed Gauge action on determinant line bundle and change of metric

        Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $?2πi\Omega$ ...
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        267 views

        Objects with trivial automorphism group

        Suppose I am working with a category of objects such that each object $X$ in the category has as a trivial automorphism group. An example of such an object would be genus zero curves with $n$-...
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        725 views

        Applications of derived categories to “Traditional Algebraic Geometry”

        I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical ...
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        135 views

        lemma II.2.4 in Harris-Taylor (about drinfeld-katz-mazur level structure on 1-dimensional $p$-divisible groups)

        Lemma II.2.4 on page 82 in Harris and Taylor's "The Geometry and Cohomology of Some Simple Shimura Varieties" (or lemma 3.2 here), says that given a Drinfeld(-Katz-Mazur) level structure $\alpha:(p^{-...
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        votes
        1answer
        176 views

        Motivation for using etale topology in representability of functors problems

        I am reading a paper that proves the representability for certain functors whose domain is the category of superschemes. The paper claims that to prove representability of functors (or possibly just ...
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        Mapping Class Group and Triangulations

        I am a physicist who's getting started with Mapping Class Group for Riemann surfaces, pants decompositions and triangulations so I apologise in advance if the following is a stupid question/wrong. I ...
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        1answer
        324 views

        Moduli space of flat connections of Lie group over a 2-torus

        We know that the moduli space of SU($N$) principal bundle's flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\rm flat}=\mathbb{E} / {S}_N = \...
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        2answers
        376 views

        Moduli space of curves

        Let $(M,\omega)$ be a symplectic manifold, and let $\mathscr{J}$ be the set of compatible almost complex structures on $M$.Finally let $A \in H^2(M,\mathbb{Z})$. Then we can consider the moduli space ...
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        SL(2,R) invariant which are not SL(2,C) invariants

        Consider four points, $\sigma_i$ i=1,2,3,4 on the line $\mathrm{Im}(z) = 0$ in the complex plane $\mathbb{C}$. Does it exist a rational function of these four points which is $\mathrm{SL}(2,\mathbb{R})...
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        66 views

        Decorated Teichmuller space of a punctured disk and moduli space of the annuls

        The decorated Teichmuller space of a disk with n punctures on the boundary and one in the interior is the the space of hyperbolic metrics on such a surface with an extra marking of an horocycle at ...

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