Questions tagged [geometric-invariant-theory]

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

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4 votes
1 answer
159 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 ...
4 votes
0 answers
216 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{...
1 vote
1 answer
238 views

Secondary fan and KN strata

Let $\mathbb{G}_m^r$ act on the affine space $\mathbb{A}^n$ through an embedding into the open dense torus. Is there a way to calculate the 1-parameter subgroups that determine the KN strata from the ...
0 votes
0 answers
67 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 ...
4 votes
1 answer
168 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 $$\...
1 vote
0 answers
135 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 $\operatorname{Proj}$...
1 vote
0 answers
230 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 ...
3 votes
0 answers
120 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}...
4 votes
0 answers
185 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 = \...
3 votes
0 answers
76 views

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 ...
1 vote
0 answers
102 views

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 ...
3 votes
0 answers
250 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 ...
8 votes
2 answers
1k views

Affine GIT is an open map?

Let $k$ be a field, $X= \text{Spec}\,A$ be an affine scheme, with $A$ a finitely generated $k$-algebra. $G=\text{Spec}\,R$ is a linearly reductive group acting rationally on A, i.e. every element of $...
2 votes
0 answers
123 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 $...
5 votes
2 answers
412 views

Affinization of $T^*\mathbb{C}P^n$

Is there an elementary description of the affinization of the algebraic cotangent bundle of $\mathbb CP^n$? I know that it can be described as some sort coadjoint orbits, but I am interested in a ...
2 votes
0 answers
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 ...
2 votes
0 answers
157 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)=...
1 vote
0 answers
105 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 ...
3 votes
2 answers
163 views

Are irregular points of an action necessarily in the closure of a larger orbit?

Suppose $G$ is an affine algebraic group acting linearly on a vector space $V$. A point $v \in V$ is stable if the orbit $Gv$ is closed and $v$ is regular (the dimension of the stabilizer of $v$ is ...
3 votes
2 answers
212 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 $W$-...
2 votes
0 answers
31 views

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 ...
7 votes
1 answer
675 views

GIT and singularities

Let $G$ be a complex reductive group acting on a complex affine variety $X$ and let $X // G = \operatorname{Spec}\mathbb{C}[X]^G$ be the GIT quotient. Is there a relationship between the singular ...
1 vote
0 answers
83 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 $...
3 votes
1 answer
117 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 ...
20 votes
10 answers
6k views

Resources on invariant theory

What are resources on invariant theory? Basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd prefer online / freeish ...
3 votes
0 answers
302 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,...
1 vote
0 answers
130 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]^{...
2 votes
0 answers
158 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 $U$...
3 votes
0 answers
207 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 ...
1 vote
0 answers
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 ...
8 votes
1 answer
552 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....
1 vote
0 answers
206 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)....
1 vote
1 answer
144 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 $...
2 votes
1 answer
172 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 ...
1 vote
0 answers
122 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 ...
5 votes
0 answers
347 views

What representation theoretic properties does the semi-invariant ring tell us?

I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)? I have been studying about semi-...
1 vote
0 answers
87 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}(...
1 vote
0 answers
73 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 ...
7 votes
1 answer
572 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 ...
2 votes
1 answer
542 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 ...
1 vote
0 answers
271 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}\...
4 votes
1 answer
234 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, ...
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 $...
2 votes
0 answers
95 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 ...
1 vote
0 answers
101 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,...
6 votes
0 answers
495 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 ...
0 votes
0 answers
146 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 ...
4 votes
0 answers
144 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 ...
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 $(\mathbb{P^1})^n$ ...
3 votes
0 answers
142 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 $\operatorname{GL}_n(\mathbb{C})$, ...

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