Questions tagged [moduli-space]
A Moduli space is a space in algebraic geometry whose points are geometric objects or isomorphism classes of these kinds of objects.
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How to prove that the moduli space $\mathcal{M}_g$ is Hausdorff using Kuranishi families?
Let $ C $ be a compact connected Riemann surface of genus $ g > 1 $. I used the definition of a Kuranishi family of $ C $ as in [Teichmüller space via Kuranishi families] 1.
Using these Kuranishi ...
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How to prove two vector bundles on $\overline{\mathcal{M}}_{g,n}$ are isomorphic without constructing an explicit isomorphism?
Question: how can one prove two vector bundles on $\overline{\mathcal{M}}_{g,n}$ are isomorphic without constructing an explicit isomorphism?
For line bundles, this can be done by computing their ...
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How to correspond extension class $\mathrm{Ext}$?
Thare are questions when I read Daniel Huybrechts' book: The geometry of moduli space of sheaves, page 37, Example 2.2.12. The content I have just copied.
Example 2.1.12 — Let $F_1$ and $F_2$ be ...
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Why integral geometric fibers of $f:X\to S$ is necessary in existence of Picard schemes?
I am reading the construction of Picard scheme in FGA explained. Now I am confused by the condition that geometric fibers of $f:X\to S$ are integral. I can not find where the proof of the main theorem ...
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Are stable hyperelliptic curves necessarily admissible double covers?
In Prop 2.4 of this paper, the authors showed an isomorphism between $\overline{\mathbf{H}}_{2,g}$ the moduli space of admissible double covers of stable $(2g + 2)$−marked curves of genus $0$ to $\...
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Coordinates for the moduli space of affine equivalence classes of $n$-gons
I investigating a certain dynamical system on the space of convex $n$-gons up to affine invariance. I would like to be able to express it in coordinates, and further to tell whether one polygon is, in ...
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The orbit of smooth hypersurfaces is closed under PGL(n+1) action
I am reading Mumford's 'Geometric Invariant Theory', where in chapter 4.2, he discussed the PGL(n+1) action on the degree d hypersurfaces in the projective space $P^n$, such hypersurfaces are ...
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Parameterization of Degree r Subline Bundles of Algebraic Vector Bundles and the Naturalness of Induced Map
For an algebraic vector bundle $V$ on an algebraic curve $C$, can one use an algebraic variety $\Sigma_V$ to parameterize all degree $r$ algebraic subline bundles of $V$? Furthermore, does this ...
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Any saturated subsheaf of a locally free sheaf is again locally free
I am reading Daniel Huybrechts's The Geometry of moduli spaces of sheaves. In the introduction of chapter 5. He uses the following result:
Proposition: Any saturated subsheaf of a locally free sheaf ...
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On a lemma in Mukai's book (convolution product relation in SL(2))
I am reading "An introduction to invariants and moduli" by Mukai and I cannot make sense of Lemma 4.57. The book goes back quite a bit in its notation, so it is not feasible for me to write ...
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Trouble calculating Weil-Peterson volumes recursively
I am having some trouble calculating Weil-Peterson volumes using Mirzakhani's recursion, following [1]. Taking $V_{0,1}(L1)$ = 0, $V_{1,2} (L1, L2) = 0$ and $V_{0,3}(L1, L2, L3) = 1$ as the base cases,...
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Gromov-Witten invariants for rational curves in $\mathbb{P}^n$
It is well known that the number or rational plane curves $C_d$ of degree $d$ passing through $3d-1$ general points of $\mathbb{P}^2$ can be computed (recursively) using the machinery of Gromov-Witten ...
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The isomorphism between Hilbert scheme of n points and moduli space
I've been reading the isomorphism between Hilbert scheme of n points and the moduli space $M(1,0,1-n)$, mainly following Huybrechts's book Lectures on K3 Surfaces, Chapter 10. On Page 215, he stated ...
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Hecke operators are multiplicative on Jacobian
I am a total beginner in this field (complex analytic geometry?), so apologies if the question is obvious. Not sure if the context is relevant, but I am studying the Eichler-Shimura construction for ...
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(co)limit diagram of derived stacks can be checked fiberwise
In the category of stacks, the (co)limit diagram can be checked by taking the fiber, then the (co)limit condition reduces to a diagram in the 2-category $Grpd$.
I am not very familiar with the derived ...
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Tangent space to locus of varieties containing a given subvariety (Hilbert schemes)
Suppose that I have a variety $X$ in $\mathbb{P}^r$ of a given type. Then I know that the tangent space to the irreducible component $\mathcal{H}_X$ of the Hilbert scheme containing $[X]$ can be ...
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Understanding Gauss-Manin connection on Arbarello's book
I am trying to understand the Gauss-Manin conecction in order to understand the definition of the Period Mapping on the Moduli space of algebraic curves of genus $g$ and its extension to the ...
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Geometric point - definition of Hilbert Scheme
I am trying to fully understand construction around Hilbert Schemes. In paper
Stein Arild Strømme: Elementary Introduction to Representable Functors and Hilbert Schemes, Parameter Spaces, Banach ...
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Zariski tangent space to a moduli space
I’m reading the paper 13/2 Ways of Counting Curves by Pandharipande and Thomas. I’m very confused with the following statement on page 8
$\S$ Deformation theory. We return now to the deformation ...
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Why is this construction of an affine curve not uniformization?
I'm learning the Shimura curve. When I reading the note of Pete L. Clark (SC2-Fuchsian.pdf (uga.edu)), I was stuck on a thinking question.
First, there is a theorem (Uniformization Theorem) about ...
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Prestack of coherent sheaves
Alper in his textbook Stacks and moduli defines the prestack of coherent sheaves over a smooth projective curve:
Let C be a fixed smooth, connected, and projective curve over an
algebraically closed ...
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What is the global complex moduli space for dimensions higher than 1?
I am trying to read the book 'Mirror Symmetry and Algebraic Geometry' by D. Cox and S. Katz. In the book it claims that 'the space of all complex structures on a given manifold $V$ is a well known ...
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On Igusa Congruence Groups $\Gamma_g(n,2n)$ and moduli interpretation of $\mathcal{A}_g(n,2n)$
Consider the congruence subgroup $\Gamma_g(n)=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\in Sp_{2g}(\mathbb{Z}):\begin{bmatrix}a&b\\c&d\end{bmatrix}\equiv\begin{bmatrix}1_g&0\\0&...
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What was the difficulty in enumerative geometry problems before physics?
I have read the book 'Enumerative Geometry and String Theory' by Katz, and it left me with some questions. It is outlined in the text how ideas from String theory and TQFT has enriched enumerative ...
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What do we know about moduli spaces of sheaves on $\mathbb{P}^n$?
I want to know some examples of moduli schemes of (geometrically) stable sheaves over a higher dimensional base scheme.
The simpliest base schemes are the projective spaces $\mathbb{P}^n$ for $n\geq2$....
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