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Results tagged with moduli-spaces
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user 4790
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.
6
votes
Accepted
Is $H^i(\mathcal{M}_g,F)$ necessarily finite dimensional for a coherent sheaf $F$?
Let $X$ be a separated Deligne-Mumford stack of finite type over a field of characteristic 0. Let $\pi\colon X \to M$ be the moduli space; assume that $M$ is quasi-projective. I claim that if $\mathrm …
- 27k
12
votes
Accepted
Rigidification and good moduli space (morphism) in the sense of Alper
It is certainly not true that $\mathcal X \to \mathcal X^H$ is a good moduli morphism, unless $H$ is linearly reductive, because when you push forward the cohomology of $H$ will come into play.
On th …
- 27k
10
votes
Accepted
Moduli Space of Abelian Varieties with a N-torsion point
No, there are no problems. The stack of principally polarized abelian varieties $\mathcal A_g$ has a universal family $\mathcal X_g \to \mathcal A_g$, which is a relative group scheme. As such, it has …
- 27k
4
votes
Accepted
Representability of Hom-sheaves of various moduli spaces
For the examples you mention, this boils down to representability of Hom sheaves of flat finitely presented proper schemes, which is due to Grothendieck. These Hom sheaves are not of finite type, thou …
- 27k
11
votes
Accepted
Are representations of a linearly reductive group discretely parameterized?
I don't think this is literally true. For example, suppose that $G$ is a finite cyclic group of order 2 generated by $s$, suppose that $L$ is an non-trivial 2-torsion invertible $A$-module over a Dede …
- 27k
9
votes
Proposition 3.93 of Harris-Morrison (rational classes on Deligne-Mumford moduli stack vs. ra...
It does generalize to Chow classes and to (classical or étale) rational cohomology. For Chow classes this is part of the basics of intersection theory of stacks, for example, in my paper Intersection …
- 27k
9
votes
Accepted
What's a good dense open of $\bar{M}_g,n(X,\beta)$?
If $X$ is convex and $g = 0$, then you can take smooth curves with distinct marked point, this will be dense. However, in general the locus of smooth curves is not dense. An easy example is that of de …
- 27k
8
votes
Accepted
Are the arithmetic genera of Cohen-Macaulay curves in a fixed homology class bounded?
I don't think this is true. Take $X = \mathbb P^1 \times \mathbb P^2$. Let $C$ be $\mathbb P^1 \times 0$, and let $C_1$ be the first infinitesimal neighborhood of $C$. The curve $C_1$ is the relative …
- 27k