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Questions tagged [geometric-invariant-theory]

for questions on geometric invariant theory (or GIT), including stability criteria and symplectic quotients.

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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}$, ...
RedLH's user avatar
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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 ...
Dr. Evil's user avatar
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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 ...
Ji Woong Park's user avatar
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258 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{...
Dr. Evil's user avatar
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"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 ...
It'sMe's user avatar
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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 $$\...
Display Name's user avatar
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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 $\operatorname{Proj}$...
jg1896's user avatar
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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}...
Display Name's user avatar
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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 ...
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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 = \...
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Why do most eigenspaces of a Lie algebra automorphism have finitely many orbits?

I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper on the Local Langlands Conjectures (omitting the "well-known" proof). Suppose $G$ is a ...
David Schwein's user avatar
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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 ...
John Klein's user avatar
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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 ...
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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 $...
Joe Silverman's user avatar
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51 views

Normal form of this group action?

Let $d\in\mathbb{N}$. We consider the vector space $V=\mathbb{C}^2\otimes\mathbb{C}[x_0,x_1]_d$ where $\mathbb{C}[x_0,x_1]_d$ is the space of homogeneous binary forms of degree $d$. We have a natural ...
Hans's user avatar
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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)=...
It'sMe's user avatar
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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 ...
user267839's user avatar
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Kähler quotients for generic $\xi\in \mathfrak{g}^*$

In this question I intentionally omit words like "(non)compact" because I am not sure about the precise setting where this question makes sense. Let $M$ be a symplectic manifold, $G$ a Lie ...
Peter Kravchuk's user avatar
3 votes
2 answers
256 views

Describing characters of a reductive group in terms of characters of a maximal torus

Say I have a reductive complex algebraic group $G$ with maximal torus $T$ and associated Weyl group $W$. I would like to be able to say that the characters of $G$ are in bijection with the $W$-...
Henry Talbott's user avatar
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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 $...
kindasorta's user avatar
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3 votes
1 answer
123 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 ...
Display Name's user avatar
3 votes
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321 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,...
jg1896's user avatar
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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]^{...
It'sMe's user avatar
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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 $U$...
It'sMe's user avatar
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3 votes
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236 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 ...
Ruotao Yang's user avatar
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78 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 ...
It'sMe's user avatar
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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)....
It'sMe's user avatar
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646 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....
YYY's user avatar
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1 answer
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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 $...
It'sMe's user avatar
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2 votes
1 answer
177 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 ...
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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 ...
It'sMe's user avatar
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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}(...
holitinh's user avatar
1 vote
0 answers
78 views

What is the functor of points of the moduli scheme of stable sheaves?

Let $\Bbbk$ be an algebraic closed field of characteristic zero. Let $(\mathrm{Sch}/\Bbbk$ denote the category of locally Noetherian schemes. Let $B$ be a projective scheme over $\Bbbk$. Let $L$ be an ...
Display Name's user avatar
7 votes
1 answer
630 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 ...
wgabrielong's user avatar
4 votes
1 answer
248 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, ...
user267839's user avatar
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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 $...
It'sMe's user avatar
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2 votes
1 answer
545 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 ...
user267839's user avatar
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1 vote
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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}\...
user267839's user avatar
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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 ...
Display Name's user avatar
1 vote
0 answers
105 views

Exists $G$-equivariant embedding with faithful representation of $G$?

Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT,...
KKD's user avatar
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6 votes
0 answers
552 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 ...
Rio's user avatar
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0 votes
0 answers
170 views

Relationship between vector bundles and modules

THE GROTHENDIECK RING IN GEOMETRY AND TOPOLOGY - M.F. ATIYAH §1. The Grothendieck ring in homotopy theory I am going to be talking about vector bundles, i.e. fibre bundles with fibre a vector space ...
Abel 's user avatar
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1 vote
0 answers
141 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 $(\mathbb{P^1})^n$ ...
tota's user avatar
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4 votes
0 answers
155 views

Bondal-Orlov conjecture on Calabi-Yau varieties

Recently, I am trying to study the various progress made on the Bondal-Orlov conjecture: Birational Calabi-Yau varieties ⟹ Equivalent derived categories. I have started reading the paper by Bridgeland ...
Rio's user avatar
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3 votes
0 answers
148 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 $\operatorname{GL}_n(\mathbb{C})$, ...
jessetvogel's user avatar
1 vote
1 answer
172 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. ...
Fernando Peña Vázquez's user avatar
6 votes
0 answers
564 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 ...
Alex Youcis's user avatar
3 votes
1 answer
252 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 ...
Tommaso Scognamiglio's user avatar
6 votes
1 answer
366 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 $...
G. Gallego's user avatar
1 vote
0 answers
166 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 ...
Mary Susy's user avatar

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