# Questions tagged [geometric-invariant-theory]

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

211
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### 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 ...

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### 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,...

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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]^{...

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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 $U$...

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### 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 ...

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### 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 ...

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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)....

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### 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....

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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 $...

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### 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 ...

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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}(...

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### 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 ...

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### 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 ...

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### 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, ...

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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 $...

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### 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 ...

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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}\...

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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 ...

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### 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,...

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### 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 ...

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### 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 ...

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### 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 $(\mathbb{P^1})^n$ ...

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### 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 ...

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### 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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### 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. ...

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### 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 ...

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### 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 ...

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### 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 $...

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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 ...

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### In GIT, why are the semistable/unstable loci defined pointwise, instead of defining semistable/unstable subschemes?

Let $k$ be an algebraic closed charateristic zero field. Let $G$ be a reductive group over $k$ and let $X$ be a scheme (not necessarily separated) of finite type over $k$. Let $L$ be a line bundle ...

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### 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 ...

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### 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 ...

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### Nontrivial Invariants of trilinear functionals

The group $\operatorname{SL}(n_1,\mathbb{C}) \times \operatorname{SL}(n_2,\mathbb{C}) \times \operatorname{SL}(n_3,\mathbb{C})$ acts on ${\mathbb C}^{n_1} \otimes {\mathbb C}^{n_2} \otimes {\mathbb C}^...

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### 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 $\...

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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 $G=SL(n+1)$ ...

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### Nef cone of a GIT quotient

I want to know how to calculate nef cone of a GIT quotient. In particular let $X$ be a projective variety and $L$ be an ample line bundle on $X$ and $G$ be a reductive algebraic group and chosen a $G$ ...

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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 ...

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### Finding étale slices

I'm trying to understand Luna's étale slice theorem by computing some examples. The theorem is usually phrased as an existence result. I wondered if there was a natural way to figure out the slice at ...

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### Group actions on affine varieties with closed orbits

The following is motivated by a (now-deleted) MSE-question by @aglearner.
Suppose that $X\subset {\mathbb C}^n$ is an affine subvariety, equipped with the classical (Euclidean) topology. Consider the ...

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### Quotient $(V -S)/G$ is a quasi-projective variety for every closed $S \subset V$ with free $G$-action

I have a question about the statement of Remark 1.4 following on Thm 1.3 from Burt
Totaro's paper "The Chow Ring of a Classifying Space" (p. 4):
Let $G$ be a reductive group over a field $k$....

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### If algebraic group $G$ acts faithfully on a $G$-qp variety $X$, then $G$ has a Faithful projective representation

In Michel Brion's survey on Linearization of algebraic group actions
is stated in Examples 3.2.2.(iv) following claim p 17
without proof:
We fix an algebraic group $G$ over field $k$ (of arbitrary ...

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### Character variety of the free group

A classical result of Fricke--Klein--Vogt from the late 1800s implies that the character variety $\chi_\mathbb{C}$ associated to the free group $F_2$ and the algebraic group $\mathrm{SL}_2(\mathbb{C})$...

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### GIT-quotients of projective-over-affine varieties

Given an action of a reductive group on a projective-over-affine variety, what are the conditions for its GIT-quotient to be again a projective-over-affine?
There is a very nice set of slides about ...

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### Actions with finite stabilizer

Consider a affine variety $X$ over the field of the complex numbers, and an action of a reductive group $G$ on $X$ (I will consider the case of $G$ not finite, in particular $G=\mathbb{C}^*$). Reading ...

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### Variety of commuting matrices

Let $G=\operatorname{GL}(n,\mathbb{C})$ and $\mathfrak{g}=\operatorname{Mat}(n,\mathbb{C})$ and let us consider the two varieties $X,Y$ defined as $$X=\{(x,y) \in G \times G \ | \ xy=yx\} $$ and $$Y=\{...

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### 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-...

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### $G$-invariant morphism and coarse moduli spaces

Let $G$ be an algebraic group acting on $X$ (a finite type scheme on $k$).
A $G$-invariant $k$-morphism $f : X \rightarrow S$ is a map such that the following commute:
$\require{AMScd}$
\begin{CD}
G \...

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### Invariant section of a linearized sheaf

I am struggling to understand what an invariant section with respect to a linearization of a line sheaf is. In Geometric Invariant Theory, given a $k$-scheme $X$ (being $k$ an algebraically closed ...

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### Invariant ring of linear algebraic groups

Let $G$ be a connected linear algebraic group. This question concerns Hilbert's 14th Problem for the adjoint action of $G$ on itself. Let $k[G]^G$ denote the algebra of regular functions on $G$ ...