Questions tagged [geometric-invariant-theory]

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

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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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1 vote
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125 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 ...
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2 votes
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58 views

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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5 votes
1 answer
243 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 ...
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1 vote
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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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2 votes
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56 views

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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  • 101
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128 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 $\...
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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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139 views

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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10 votes
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478 views

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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  • 457
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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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2 answers
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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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0 answers
87 views

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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9 votes
2 answers
635 views

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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2 votes
0 answers
202 views

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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  • 33
3 votes
2 answers
210 views

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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18 votes
3 answers
1k views

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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5 votes
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264 views

When is the semi-invariant ring is a polynomial ring or a hypersurface?

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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2 votes
1 answer
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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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  • 35
5 votes
1 answer
333 views

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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1 vote
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241 views

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 $G$ ...
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3 votes
1 answer
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Why is the image of closed invariant subsets closed? Mumford, GIT, Theorem 1.1

I'm currently studying Mumford's Geometric Invariant Theory. Unfortunately, I'm stuck understanding a detail in Theorem 1.1. (Partial) Claim of Theorem 1.1 Let $X = \operatorname{Spec} R$ be an affine ...
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105 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 $k$ (...
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5 votes
1 answer
205 views

Invariant ideal generated by invariant elements

Let $G$ be a complex reductive group acting linearly on $\mathbb{C}^n$ and let $X$ be a $G$-invariant closed subvariety of $\mathbb{C}^n$. Is $X$ the zero-set of finitely many $G$-invariant functions? ...
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2 votes
1 answer
124 views

Geometric quotients obtained by throwing away limits

Set-up: Consider the action of $\mathbb{C}^*$ on $\mathbb{C}^4$ defined as follows: $(t,(x,y,z,w))=(tx,ty,t^{-1}z,t^{-1}w)$. I know that the affine GIT quotient is equal to $$\phi: \mathbb{C}^4 \to \...
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4 votes
1 answer
196 views

Question about valuation and blow up (a lemma in GIT book)

I'm reading Mumford's book Geometric Invariant Theory and confused about the proof of a lemma on Page 91&92: Lemma: Let $V_0$ be a smooth surface over an algebraically closed field $k$ with char$...
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1 answer
159 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 ...
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3 votes
1 answer
152 views

Is the irreducible locus of the character variety a principal bundle in Zariski topology?

Let $\Sigma$ be a compact orientable surface and let $G$ be a reductive algebraic group (say, $G=\mathrm{SL}_n(\mathbb{C})$ for simplicity). The representation variety is $$ X_G(\Sigma) = \mathrm{Hom}(...
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  • 305
2 votes
1 answer
205 views

Is there an $SL_n$-invariant functional on the space of rational functions on the projective space $\mathbb P^{n-1}$?

Let the group $SL_n$ act on the projective space $\mathbb P^{n-1}$ in the standard way (both defined over $\mathbb C$). Is there an $SL_n$-invariant (linear) functional on the space of rational ...
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  • 2,175
0 votes
1 answer
231 views

Self duality and anti-self duality of Weyl curvature in four dimension

I am trying to compute explicitly in terms of extrinsic curvature the self dual part $W^+$ and the anti-self dual part $W^-$ of the Weyl tensor $W$ associated with a codimension 1 submanifold into ...
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4 votes
0 answers
70 views

Is the union of conic orbits for a reductive group Zariski closed?

Let $G$ be a reductive group over an algebraically closed field $k$ of characteristic $p>0$. If $V$ is a rational $G$-module then we can define the Hilbert nullcone $\mathcal{N}(V)$ to be the zero ...
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3 votes
1 answer
261 views

Question on geometric invariant theory

I'm reading Geometric Invariant Theory by Mumford, and confuse about the Proposition 2.4 on P54. It states that: Let $G$ be a reductive group, act on an algebraic scheme. Then the action of $G$ on $X$ ...
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0 votes
0 answers
107 views

Equivalence between coactions and actions plus a linearization line bundle

Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all ...
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  • 317
2 votes
0 answers
101 views

Determining a toric GIT quotient

Consider the following $G = (\mathbb{C}^*)^{\times 3}$-action on $\mathbb{A}^6$: $(\lambda_0,\lambda_1,\lambda_2) \cdot (x_0,x_1,x_2,y_0,y_1,y_2) = (\lambda_0x_0,\lambda_1x_1,\lambda_2x_2,\frac{\...
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  • 197
4 votes
0 answers
94 views

Cover by $K$-invariant affine open sets

Let $X$ be a non-singular complex algebraic variety (quasi-projective if necessary) and $K$ a connected compact Lie group acting on $X$ real algebraically, i.e. the action map $K \times X \to X$ is ...
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7 votes
1 answer
491 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 ...
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7 votes
1 answer
249 views

An explicit negative solution to the Lüroth problem for non-algebraically closed fields

Let $\mathsf{k}$ be a field of characteristic $0$, and consider $\mathsf{k}(x,y)$. If $\mathsf{k}$ is algebraically closed, then every field $L$ such that the inclusion $\mathsf{k} \subset L \subset \...
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5 votes
0 answers
178 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 ...
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  • 1,713
2 votes
1 answer
206 views

Jordan decomposition on the dual Lie algebra

$\newcommand\fg{\mathfrak g}\newcommand\gl{\mathfrak{gl}}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field, and let $G$ be a smooth, affine algebraic ...
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  • 7,333
12 votes
1 answer
326 views

Moduli of smooth curves in $|\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2,2)| $ and their invariants

It is well known that any smooth curve $C\in |\mathcal{O}_{\mathbf{P}^1\times\mathbf{P}^1}(2,2)| $ has geometric genus equal to 1, so its isomorphism class is determined by its $j$-invariant. ...
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3 votes
0 answers
86 views

How big is the complement of stable locus $\operatorname{Bun}G$

Let $\Sigma$ be a smooth projective curve, and $G$ a reductive group. Let $\operatorname{Bun}G$ be the stack of principal $G$ bundles on $\Sigma$ (with a fixed topological type). What is the ...
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1 vote
0 answers
120 views

Descent of projective bundles

A problem studied in GIT is the descending of vector bundles (or more in general coherent sheaves) to quotients. It is a result of Kempf that whenever we have a vector bundle over a quasiprojective ...
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  • 287
4 votes
0 answers
193 views

Quotients of toric varieties

This is a follow up of this question. Given a toric variety $X$ with a fan $\Sigma$ and a finite group $G$ acting on $X$, we know that the GIT quotient $X/G$ exists. However, as stated in the answer ...
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2 votes
0 answers
172 views

Fiber product arising from reductive group action on varieties

Let $G$ be a reductive group acting on a smooth projective $X$. Let $P$ be a parabolic subgroup of $G$, and $Y$ a locally closed subvariety invariant under $P$. Assume in addition $Y$ is smooth. ( ...
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3 votes
1 answer
148 views

(Co)tangent sheaves to good quotients

Suppose given a variety $X$ over an algebraically closed field $K$, $\mathrm{char}K = 0$, equipped with an action of a reductive group $G$. Suppose also that $X$ admits a good quotient $p\colon X\to Y:...
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3 votes
0 answers
147 views

Lifting of curves in characteristic zero

Let $K$ be an algebraically closed field of characteristic zero. Let $G$ be an affine reductive group over $K$, and let $H$ be a closed reductive subgroup of $G$. Since $H$ is reductive the GIT ...
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  • 4,562
5 votes
1 answer
352 views

Fibre of GIT morphism

Let $ V $ be an affine variety (over $ \mathbb C$) with an action of a reductive group $ G$. I would like to consider the morphism $$ \pi : V \rightarrow V // G = Spec \, \mathbb C[V]^G $$ Let $ v \...
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0 votes
2 answers
350 views

Motivating the Quotient of an Algebraic Variety

Let $X$ be a variety with a $G$-action by an algebraic group. My question refers to a motivating example from: https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf Here is the relevant ...
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  • 679
1 vote
1 answer
179 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 ...
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