All Questions
Tagged with geometric-invariant-theory ag.algebraic-geometry
195 questions
4
votes
1
answer
245
views
Group action on affine variety induces faithful action on tangent space
I have a queestion about the proof of Lemma 2.2 from the paper arxiv 1105.3739:
Let $G$ be a group acting faithfully on an irreducible affine variety $X=\operatorname{Spec}(A)$ over $k= \Bbb C$. ...
- 5,966
5
votes
4
answers
1k
views
Stable points in GIT: geometric picture
Is there a geometric picture justifying why "stable points" in GIT (Geometric Invariant Theory) are actually called "stable"? Stable, with respect to which effect? (Here, I have ...
- 619
3
votes
0
answers
126
views
Parametrization of indecomposable modules via quiver varieties
Let $k$ be an algebraically closed field, $Q$ a quiver without oriented cycles and $m^\alpha (Q)$ the variety of quiver representations with dimension vector $\alpha$. There is a canonical algebraic ...
- 696
3
votes
1
answer
320
views
Is the Hilbert Mumford Criterion true over the reals?
The Hilbert Mumford Criterion as in Wallach Theorem 3.24 says:
Let $G$ be a linearly reductive subgroup of $GL(n, \mathbb{C})$. Let $(\sigma, V)$ be a regular representation of $G$.
For a vector $v \...
- 161
1
vote
0
answers
138
views
Quotients of open subsets of the semi-stable locus
This is a rewrite of a deleted question. I've decided to focus on one particular example mentioned in that question. Below a point means a closed point.
Let $U$ be the set of irreducible non-cuspidal ...
- 23.5k
2
votes
1
answer
143
views
$G$- Fixed Point Scheme explicitly
Let $G$ be an abstract finite group acting on a separated $k$-scheme $X$. ($k$ a field; note we can canonically promote $G$ to a $k$- scheme). Then a result by Demazure and Grothendieck (in "...
- 5,966
1
vote
0
answers
76
views
Seeking for bridges to connect K-stability and GIT-stability
We consider the variety $\Sigma_{m}$ := {($p$, $X$) : $X$ is a degree $n$ + 1 hypersurface over $\mathbb{C}$ with mult$_{p}(X) \geq m$} $\subseteq$ $\mathbb{P}$$^{n}$ $\times$ $\mathbb{P}$$^{N}$, ...
- 41
2
votes
0
answers
182
views
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
1
answer
253
views
Symplectic structure of Higgs branch
I've been reading Kamnitzer's survey Symplectic resolutions, symplectic duality, and Coulomb branches. Here the Higgs branch is defined as a projective GIT quotient, but I couldn't figure out how the ...
- 323
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
0
votes
0
answers
71
views
"Approximating" ring of semi-invariants
I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...
- 839
4
votes
1
answer
186
views
Are the two notions of free $\mathbb{G}_a$-actions equivalent?
Consider a finitely generated integral $\mathbb{C}$-domain $B$. An algebraic $\mathbb{G}_a$-action on $X:=\mathrm{Spec}(\mathcal{O}(X))$ is equivalent to a locally nilpotent $\mathbb{C}$-derivation
$$\...
- 930
1
vote
0
answers
150
views
There exists noncommutative geometric invariant theory?
In this question, I am going to consider noncommutative projective algebraic geometry, as introduced by Artin and Zhang in the seminal paper Noncommutative projective schemes. The $\operatorname{Proj}$...
- 3,318
3
votes
1
answer
206
views
Are there geometric $\mathbb{G}_a$-quotients with trivial stabilizers, not being principal bundles?
Consider algebraic $\mathbb{C}$-schemes. The group scheme $\mathbb{G}_a$ is the scheme $\mathbb{A}^1$ with the addition. This is not a reductive group. Here I want to know some examples of $\mathbb{G}...
- 930
1
vote
0
answers
257
views
Confusion regarding the invariant rational functions
I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below)
It says that "every invariant rational function can ...
- 839
4
votes
0
answers
227
views
Elementary proof Hilbert-Mumford stability criterion for $\operatorname{GL}_n(\mathbb{C})$
In An elementary proof of the Hilbert-Mumford criterion, B. Sury gives an elementary proof of the Hilbert-Mumford semi-stability criterion for $G = \operatorname{GL}_n(\mathbb{C})$ (and $G = \...
- 7,320
3
votes
0
answers
267
views
Does the orbit in geometric invariant theory have natural scheme structure
Let $X$ be a scheme locally of finite type over a sufficiently "nice" base scheme $S$ (nice in sense of reasonable "finiteness conditions", for sake of simplicity let's start as ...
- 5,966
2
votes
0
answers
137
views
Is the GIT quotient of a finite map of varieties again a finite map?
Let $K$ be an algebraically closed field of characteristic $0$, let $X/K$ and $Y/K$ be quasi-projective varieties, and let $f:X\to Y$ be a morphism. Let $G/K$ be a reductive group that acts stably on $...
- 47.4k
2
votes
0
answers
188
views
Help with Macaulay2 computation of invariant ring
Consider the algebraic group $G:=\operatorname{SL}_{2}\times\operatorname{SL}_{2}$ acting on $V:=\operatorname{Mat}_{2\times 2}\oplus\operatorname{Mat}_{2\times 2}$ via the action $(A,B)\,\cdot\,(X,Y)=...
- 839
1
vote
0
answers
119
views
Action by finite abstract group on affine scheme
Let $X:=\operatorname{Spec}(R)$ an affine Noetherian scheme and $G$ a finite group acting on $X$. Then it is known that the quotient $Y=X/G$ exists as affine scheme $\operatorname{Spec}(R^G)$, let set ...
- 5,966
1
vote
0
answers
91
views
Quotient sheaf obtained from a quotient of $\mathrm{SL}_2$ having a section
$\DeclareMathOperator\SL{SL}$Let $K$ be a field and let $G$ be a $K$-defined, closed, algebraic subgroup of $\SL_2$. Denote by $\mathcal{C}$ the site whose objects are $K$-schemes, with your favorite $...
- 2,907
3
votes
1
answer
131
views
Finer classification of semistable sheaves
Usually in the moduli space of semistable sheaves, two semistable sheaves correspond to one point if and only if they are S-equvialent, i.e. the graded objects associated to their Jordan-Holder ...
- 930
3
votes
0
answers
345
views
On Noether's Problem
Noether's problem is a famous problem in invariant theory, introduced in the 1910's by Emmy Noether in relation to the inverse Galois problem. It is as follows:
Noether's Problem: Let $F=k(x_1,\dotsc,...
- 3,318
1
vote
0
answers
156
views
Software for computing invariant rings
I have an linearly reductive algebraic group $G$ acting regularly on an affine variety $X$(over an algebraically closed field of characteristic 0). I want to compute the invariant ring $\mathbb{K}[X]^{...
- 839
2
votes
0
answers
174
views
How are tangent spaces related via geometric quotient?
Let $G$ be a linearly reductive group acting regularly on an irreducible affine variety $X$ (over an algebraically closed field of characteristic zero). Suppose there's a $G$-stable open subvariety $U$...
- 839
3
votes
0
answers
287
views
When a stack quotient coincides with GIT quotient?
Let $G$ be a reductive group over $\mathbb{C}$, and $H=H_r\ltimes H_u$ be a subgroup of $G$. Here, $H_u$ is unipotent and $H_r$ is reductive.
Question: Is it true that when $G/H$ is open in its affine ...
- 31
1
vote
0
answers
80
views
When is $Y$ not an orbit closure?
Let $G$ be a linearly reductive algebraic group acting regularly on an affine space over $\mathbb{A}^n$ an algebraically closed field $\mathbb{K}$. Let $Y$ be a $G$-invariant (closed) affine ...
- 839
1
vote
0
answers
208
views
Is $\langle\chi,\lambda\rangle=0$, whenever the limit exists? Where is the mistake?
Suppose $G$ is a linearly reductive algebraic group acting linearly on a finite dimensional vector space $V$ over $\mathbb{C}$. This induces an action on the coordinate ring $\mathbb{C}[V]$ (see here)....
- 839
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....
- 197
1
vote
1
answer
155
views
Invariant ring of the subvariety
Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also $...
- 839
2
votes
1
answer
187
views
Orbits in the open set given by Rosenlicht's result
Let $G$ be a linearly reductive algebraic group, and let $X$ be an irreducible affine variety, over an algebraically closed field $\mathbb{K}$, with a regular action of $G$. By Rosenlicht's result, we ...
- 839
1
vote
0
answers
127
views
How to determine if an invariant rational function is defined at the $\theta$-polystable point
Background:
Let $A$ be a finite-dimensional (associative and unital) algebra over $\mathbb{C}$. Assume there is a quiver $Q=(Q_0,Q_1)$, where $Q_0$ is the set of vertices and $Q_1$ is the set of ...
- 839
1
vote
0
answers
93
views
Sufficient condition for moduli space of slope-stable bundles to be non-empty
I'm trying to find to a statement concerning the non-emptiness of the moduli space of slope stable vector bundles over a Kähler surface in the literature.
Let $X$ be a Kähler surface. Let $\mathscr{M}(...
- 11
7
votes
1
answer
835
views
Intuition for Luna's Étale Slice Theorem
I am trying to understand the intuition for Luna's Étale Slice Theorem in the affine setting over $\mathbb{C}$.
Here is the setup. Let $X$ be an affine algebraic variety and $G$ a reductive group ...
- 83
4
votes
1
answer
255
views
Example of a line bundle not admitting a $\operatorname{PGL}(n+1)$-linearization in Mumford's GIT
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Proj{Proj}\DeclareMathOperator\Pic{Pic}$I have a question about an example for a line bundle not admitting a
$G$-linearization from Mumford's GIT, ...
- 5,966
1
vote
0
answers
95
views
Is $U\subseteq X^{s}(L)$?
Let $X$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristics $0$. Let $G$ be a connected, linearly reductive, affine algebraic group acting regularly on $...
- 839
2
votes
1
answer
563
views
Proposition 1.5 in Mumford's Geometric Invariant Theory
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\pr{pr}$I have some problems to understand the proof of Proposition 1.5 from Mumford's ...
- 5,966
1
vote
0
answers
275
views
Corollary 1.6 in Mumford's Geometric Invariant Theory
I do not understand fully the proof of Corollary 1.6 from Mumford's Geometric Invariant Theory (§3: Linearization of an invertible sheaf, p 35):
Corollary 1.6
$\DeclareMathOperator\Spec{Spec}\...
- 5,966
2
votes
0
answers
98
views
What is the natural linearization on differentials?
Let $\Bbbk$ be a field. Let $G$ be an affine algebraic group over $\Bbbk$. Let $X$ be a scheme over $\Bbbk$. Let $G$ act on $X$ with the action morphism $\sigma:G\times X\to X$. There are two ...
- 930
6
votes
0
answers
679
views
Learning about moduli spaces of sheaves
I am a Ph.D. student and starting a side project with a fellow student on Moduli spaces. Our plan was to start with the book on Invariants and Moduli by Mukai (starting from chapter 5) and use the ...
- 345
1
vote
0
answers
147
views
Question regarding Hilbert scheme of points
$\DeclareMathOperator\SL{SL}$Let us consider $\SL(2,\mathbb{C})$ quotients of $(\mathbb{P^1})^n$ in the following sense. We consider diagonal action of $\SL(2,\mathbb{C})$ over $(\mathbb{P^1})^n$ ...
- 585
3
votes
0
answers
164
views
Class of finite quotient affine space in Grothendieck ring of varieties
Let $G$ be a finite group acting linearly on affine space $\mathbb{A}^n_k$ over $k = \mathbb{C}$. Since the action is linear, it can be extended to an action of $\operatorname{GL}_n(\mathbb{C})$, ...
- 197
1
vote
1
answer
186
views
On the trivialization of the sheaf of kahler differentials on the G-invariant topology
Let $X$ be a connected, smooth affine algebraic variety over an algebraically closed field $K$ of characteristic zero. Assume we have a finite group $G$ acting on $X$ by morphisms of $K$-schemes. ...
- 541
6
votes
0
answers
591
views
Affine GIT quotients and the excursion algebra in Fargues–Scholze
Some background:
Let us fix a non-archimedean local field $E$ with residue characteristic $p$, and let $G$ be some connected reductive group over $E$. In [FS, §VIII.1.1] the authors define a moduli ...
- 787
3
votes
1
answer
260
views
Invariants of general linear groups under torus action
Let $G=GL_n$ be the general linear group (let's say over an algebraically closed field of char $=0$). Let's denote as $T$ the torus of diagonal matrices: is there an explicit description of the ...
- 1,061
7
votes
1
answer
397
views
Is there a Chevalley map for spherical varieties?
If $G$ is a reductive group, $T$ a maximal torus and $W$ its Weyl group the Chevalley restriction theorem (in its "multiplicative" version) gives an isomorphism between the GIT quotient of $...
- 553
1
vote
0
answers
176
views
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 ...
- 27
5
votes
1
answer
343
views
About closed points in symmetric product schemes over a finite field
Let $k=\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation):
Let $N$ be a positive ...
- 1,906
1
vote
0
answers
144
views
Non-empty stable locus of an irreducible component
I have a vague question:
Let $X$ be an algebraic pre-scheme and $G$ be a linear reductive group. Consider the G.I.T. quotient $X{/\!/}G$. Is there any result (maybe in some special case) which tells ...
- 839
4
votes
1
answer
257
views
Question regarding semistability of a point of GIT quotient
$\DeclareMathOperator\SL{SL}$I am currently looking at the paper titled "$\SL(2,\mathbb{C})$ quotients de $(\mathbb{P^1})^n$" by Marzia Polito. The author has considered diagonal action of $\...
- 585