All Questions
Tagged with geometric-invariant-theory reference-request
16 questions
2
votes
0
answers
182
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GIT quotient and orbifolds
Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...
- 2,751
4
votes
0
answers
310
views
GIT quotient of a reductive Lie algebra by the maximal torus
Let $G$ be a connected complex reductive group with Lie algebra $\mathfrak{g}$. One knows a lot about the GIT quotient $\mathfrak{g}/\!/G$: the invariant ring is a free polynomial algebra on $\mathrm{...
- 2,751
1
vote
0
answers
108
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Iterated quotients in GIT
Suppose that $G$ is a reductive group that acts algebraically on an affine variety $X$ over an algebraically closed field $k$.
Suppose also that $G$ is equipped with a normal abelian subgroup $N$ such ...
- 18.9k
11
votes
1
answer
938
views
An easy textbook for geometric invariant theory and moduli space which makes use of scheme theory
I would like to study geometric invariant theory and moduli theory.
It seems that a standard textbook for these fields is "Geometric Invariant Theory" written by D.Mumford, J.Fogarty and F....
- 4,111
1
vote
0
answers
176
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When the action of reductive group on algebraic variety is not equidimensional?
I saw the question When is an almost geometric quotient flat? which said
"The quotient $\pi$ is flat if and only if $\pi$ is equidimensional and $X$ is smooth".
I am curious is there an ...
- 63.9k
8
votes
0
answers
235
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Stability of nodal hypersurfaces
We denote by $\Pi_{n,d}$ the space of homogeneous polynomials of degree $d$ in $n+1$ variables $x_0,\ldots,x_n$, i.e. $\Pi_{n,d}=\Gamma(\mathbb{P}^n(\mathbb{C}),\mathcal{O}(d))$. The group $G=SL(n+1)$ ...
11
votes
2
answers
684
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Invariants of $\mathrm{GL}_n$ representations
$\DeclareMathOperator\GL{GL}$Let $V=\mathbb C^n$ be the natural representation of $\GL_n(\mathbb C)$ and let $W=\operatorname{Sym}^2(V)$ be the symmetric square representation. Let $W^k$ denote the ...
- 63.9k
3
votes
0
answers
147
views
A good stratification of a variety on which an algebraic group acts
Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0
(a reduced separated scheme of finite type over $k$).
Let $G$ be a connected linear algebraic group over $k$ (...
- 14.2k
4
votes
1
answer
278
views
Is quotient of projective scheme over arbitrary base by a finite group also projective
This question probably follows from standard geometric invariant theory. If true I'd to know a reference for it.
Given a projective scheme $X\rightarrow S$ over the base $S$. Let's assume a finite ...
- 8,529
5
votes
0
answers
245
views
Pseudoreflection groups in affine varieties
Suppose $\mathsf{k}$ is an algebraically closed field of zero characteristic. Chevalley-Shephard-Todd (C-S-T) Theorem in one of its equivalent versions is the following result:
(C-S-T): Let $G$ be a ...
- 3,318
6
votes
1
answer
2k
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Preparation for GIT (Geometric Invariant Theory)
I am trying to read Mumford's Geometric Invariant Theory, however, I find my knowledge in algebraic geometry is inadequate. My knowledge is at the level of Hartshorne's Algebraic Geometry. Mumford ...
- 8,529
2
votes
1
answer
95
views
Reference on reductive group acting on quotient algebra
In unpublished notes by Yi Hu (which appear to be no longer online), I found the following:
Corollary 2.4.5. Let the characteristic of $k$ is zero. Assume that a reductive group $G$ acts rationally ...
- 21
3
votes
1
answer
619
views
When is an almost geometric quotient flat?
All varieties here are over $\Bbb C$. Let $G$ be a reductive algebraic group acting algebraically on affine $n$-space $\Bbb A^n$. Let $R$ be the coordinate ring of $\Bbb A^n$. Assume that the natural ...
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- 1
3
votes
1
answer
670
views
How does one show that slope stability of a vector bundle is an open condition with respect to the polarisation?
I would like a source for the following result, which I expect to be true (probably well known):
Let $X$ be a complex projective variety, $L$ an ample line bundle and $E$ a slope stable vector bundle ...
- 419
3
votes
2
answers
367
views
Intersection theory for $G$-varieties - an action on the chow ring?
Let $G$ be a reductive algebraic group. Let $X$ be a $G$-variety and consider any closed subvariety $Z$ of $X$. Since any $g\in G$ acts as an automorphism, we know that $g.Z$ is again a closed ...
- 1,780
1
vote
1
answer
219
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invariants of plane quartics
Does anybody know a good reference where the invariants for plane quartic curves are developed?
- 39.3k