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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Reference request: Moduli space of immersed polygons
I am looking for references describing the structure of the set of immersions of the disk $D^2$ in the Euclidean plane $\mathbb{R}^2$ or in the hyperbolic plane $\mathbb{H}^2$ such that the boundary ...
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Metrics on the Moduli / Teichmüller space of higher dimensional complex tori / abelian varieties
I am trying to study canonical metrics on the moduli / Teichmüller space of complex tori/abelian varieties. What would be a good reference for this?
Usually the classical references like Complex Tori ...
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Fine moduli space of elliptic curves(level 3). Applications?
Recently I have learned that there is a fine moduli space for elliptic curves $X$ with fixed morphism $\phi: (\mathbb{Z}/3\mathbb{Z})^2 \rightarrow X[3]$. In Dolgachev's "Classical Algebraic ...
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Isomorphism types of curves in the second Hirzebruch surface
I am not well versed in algebraic geometry but am trying to learn about certain linear systems over the 2nd Hirzebruch surface. I would appreciate any references to where I can learn about this kind ...
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Do set-theoretical freeness of an additive $\mathbb C^+$ group action + geometric quotient imply properness?
Let $\mathbb G_a=\mathbb C^+$ denote the additive group scheme over $\mathbb C$. Suppose we have an affine complex variety $X$ and a set-theoretically free $\mathbb G_a$ action on $X$ that admits a ...
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Yang-Mills theory and... reciprocity laws!
This question is about two papers which I do not understand. Ergo, most probably it does not make any sense.
I am, moreover, no number theorist; -- and, hence, would be very glad if someone more ...
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Computing the universal family for a moduli problem
Fix $n\ge 0$. I'm interested in understanding a compactification of the moduli of ordered $n$-tuples
$$(a_1,\dots,a_n)\in\mathbb R^n,\qquad a_1\ge \cdots \ge a_n,$$
taken modulo the diagonal ...
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Limit of Higgs bundles
Let $X$ be a smooth complex projective surface, let $\mathcal{M}$ be the moduli space of semistable Higgs sheaves with fixed Hilbert polinomial with respect to a polarization of $X$.
There exists a ...
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Family of vector bundles over a relative A^1
Let $A=\mathbb{C}[[s]]$, I am looking for an example of a non-trivial vector bundle (or $\operatorname{SL}_n$-torsor) on $\mathbb{P}^{1}_{A[t,t^{-1}]}$ that is trivial over $\mathbb{P}^{1}_{\mathbb{C}...
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Is it true that $\operatorname{Alb}(X \times Y ) \cong \operatorname{Alb}(X) \times \operatorname{Alb}(Y) $?
$\DeclareMathOperator\Alb{Alb}\DeclareMathOperator\Pic{Pic}$Let $X$ and $Y$ be two smooth projective varieties over $\mathbb{C}$. My question is, is it true that,
$$\Alb(X \times Y ) \cong \Alb(X) \...
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For a smooth projective curve $X$, is the map, $M(r,d)\rightarrow \mathrm{Pic}^d(X)\cong \mathrm{Pic}^0(X)$ an albanese morphism?
$\DeclareMathOperator\Pic{Pic}$Let $X$ be a smooth projective curve and $M(r,d)$ be the moduli space of semistable vector bundles of rank $r$ and degree $d$ over $X$. My aim is to show that, when we ...
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Variants of moduli spaces of Riemann surfaces
The moduli space $M_{g,n}$ of genus $g$ Riemann surfaces with $n$ marked points is well-studied and is finite-dimensional. I'm curious about the moduli space $M_{g,\vec{n}}$ of genus $g$ Riemann ...
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What is Baer's theorem?
I've been reading the paper "The global nilpotent variety is Lagrangian" by Ginzburg, which aims to show that for the stack of Higgs $G$-bundles over a curve $X$, which is isomorphic to the ...
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Stack quotient of a point vs. classifying spaces
I want to work over the site of smooth manifolds with Grothendieck topology induced by open immersion. I am primarily concerned with the the following question: in what ways can we classify vector ...
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Isomorphism between Moduli spaces of vector bundles with fixed determinants, over a curve
Let C be a smooth projective curve, and let $L_1$ and $L_2$ be two line bundles with degree $d_1$ and $d_2$ respectively, where $gcd(r,d_i)=1$ for $i=1,2$. Suppose $M(r, L_1)$ and $M(r, L_2)$ are two ...
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