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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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105 views

Moduli space of nilpotent Lie algebras

Fix a nilpotent Lie algebra $L$ over some char 0 field $k$ which is naturally graded, i. e. isomorphic to graded algebra $\bar L$ associated to lower central filtration. I'm interested in some ...
4
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0answers
34 views

Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $−2πi\Omega$ ...
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77 views

Alternatives to the ring of invariants depicting the orbit closures?

Let $G$ be an affine algebraic group over $\mathbb{C}$ and $V=\textrm{Spec}(A)$ an affine $G$-variety. Assume that $G$ is reductive. My understanding is that a central object in studying the action of ...
2
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1answer
56 views

Reference on reductive group acting on quotient algebra

In unpublished notes by Yi Hu (which appear to be no longer online), I found the following: Corollary 2.4.5. Let the characteristic of $k$ is zero. Assume that a reductive group $G$ acts rationally ...
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134 views

Spherical varieties as GIT quotients

Let $X$ be a normal projective variety with finitely generated Cox ring. Consider its characteristic space $p:\widehat{X}\rightarrow X$. This means that there is a torus $T$ acting on $\overline{X}=...
2
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1answer
79 views

Analytic sections of a GIT quotient lying in the Kempf-Ness set

I would like to understand whether the following is true. Given a complex reductive group $G$ (for me $G = \operatorname{PGL}_n(\mathbb{C})$) acting on a vector space $V$ (for me $V = M_n(\mathbb{C})^{...
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191 views

Good quotients and coarse moduli spaces

I started study the moduli space theory, by Newstead's book ''Lectures on moduli problems and orbit spaces". Theorem 3.21 says that given a variety $X$ and a line bundle $L$ over $X$, then for any L-...
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81 views

Topological criterion for GIT semistability

Let $X$ be a complex algebraic variety and $G$ a complex reductive group acting on $X$. Let $L$ be a linearization of this action, i.e. a line bundle with a linear action of $G$ covering that of $X$. ...
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106 views

Invariants and subgroups

Let $G$ be an affine algebraic group over some algebraically closed field $K$, and let $H$ be a closed subgroup. Assume that $G$ acts algebraically on an affine variety $X$. Assume that $X'\subseteq ...
9
votes
1answer
200 views

A duality result for Coxeter groups

Short version: if $G$ is a Coxeter group and $H \subset G$ is a parabolic subgroup, both acting on a space $V$, is it true that the invariant-coinvariant algebra $(S(V)_G)^H$ has a natural bilinear ...
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1answer
165 views

GIT quotients of open subsets

Let X be a projective variety on with a action of reductive group G. Let L be a G Linearised ample line bundle on X. Let U be a G stable open subset of X. Let $U^{ss}:=X^{ss}\cap U$. Is it true that $...
3
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1answer
234 views

GIT quotient vs. largest Hausdorff quotient

Let a group $G$ act on a (not necessarily irreducible) algebraic variety over ${\bf C}$. It seems to be well-known that the quotient in the sense of geometric invariant theory (i.e., the categorical ...
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170 views

Progress since Luna's theorem on smooth invariants

In 1976, Luna proved the following important theorem of smooth invariant theory: Let $G$ be a real reductive Lie group and a representation of $G$ on a real finite dimensional vector space $V$. ...
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115 views

Binary forms and equivariant derived category

One of the classical questions in invariant theory is the classification of binary forms, i.e., the description of polynomial invariants of the ${\rm SL}_2(\mathbb{C})$-action on ${\rm Sym}^d \mathbb{...
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vote
1answer
126 views

un-ordered distinct $n$-tuples of points on $\mathbb P^1$

I want to study the space $Y$ of all un-ordered $d$-tuples of points on $\mathbb P^1$. By considering the space $V_d$ of homogeneous polynomials of degree $d$ in two variables, one may identify $Y$ ...
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0answers
22 views

Orbit of a transverse manifold under the action of an algebraic group

Consider an algebraic group acting on a affine manifold. Suppose that S is an affine submanifold transverse to the action. Are there some conditions on the group, the action or S such that the orbit ...
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0answers
105 views

Generators of the same degree in a graded ring and GIT quotient

Let $S$ be a graded ring i.e. $S_d\cdot S_e \subset S_{d+e}$ where $S_d$ is the degree $d$ part of $S$. Assume $S_{<0}$ vanishes. For simplicity, you may think of $S$ as a graded subring of $\...
4
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2answers
300 views

The closure $\overline{Gx}$ for an affine variety on which an reductive algebraic group acts

Let $G$ be a reductive group acting on an affine variety $X$. For simplicity, one may assume $G=SL_n$ or $G=U_n$ and assume the field is $\mathbb C$. Given this one can show $\mathbb C[X]^G$ is ...
2
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1answer
140 views

G-sweep of irreducible sub variety

Let $G$ be a connected reductive algebraic group and $X$ be a $G$-variety. Let $Y$ be a $G$-invariant irreducible sub variety of $X$ which has non-trivial intersection with the semi stable locus $X^{...
2
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1answer
350 views

lines in projective spaces [closed]

Let $\{v_1,v_2, \cdots , v_n, w_1,w_2, \cdots ,w_n\}$ be a basis of $\mathbb C^{2n}$. For a $n$-dimensional subspace $V \in Gr(n,\mathbb C^{2n})$ define another $n$ dimensional subspace $\bar{V} \in ...
3
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1answer
270 views

Quotient of complex manifold by a free and locally proper action (difficulty with reading German)

Let $X$ be a complex manifold with an action of $G=GL(n,\mathbb{C})$ which is free and locally proper (each point of $X$ has a $G$-invariant neighborhood on which $G$ acts properly.) Satz 24 of the ...
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163 views

Ring of invariants and Borel subgroup

Let $G$ be a connected algebraic group (can assume $G$ to be reductive) acting on a $k$-algebra $A$. Let $B$ be a Borel subgroup of $G$. Q. Is it generally true that the the ring of invariants $A^...
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124 views

Injectivity of a standard map in quiver representation

Let $X$ be a smooth projective variety, and assume its divisor class group is finite and free. Let $E_1,E_2,\ldots,E_n$ be line bundles on $X$. Define $L_k=E_1+\ldots E_k$, and let $Q$ be the ...
7
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1answer
203 views

Standard Monomial basis for other types

For the algebraic group $SL_n$ (type $A_{n-1}$) and for a dominant weight $\lambda$ the standard monomials are indexed by the semi-standard young tableaux of shape $\lambda$ and they form a basis for ...
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1answer
138 views

Is the conjugation action linearizable?

Let $G$ be a reductive algebraic group over some algebraically closed field $k$. Recall that, given an algebraic variety $X$ with an action of $G$, it is said that this action is linearizable if there ...
8
votes
1answer
374 views

Interactions (functors) between equivariant sheaves for different groups?

Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity). To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is ...
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241 views

Is $\widetilde M_{0, n}$ a Mori Dream space?

I'm reading on $\overline M_{0, n}$ and $\widetilde M_{0, n}$. I know that $\overline M_{0, n}$ is a Mori Dream space for $n \leq 6$ and not a Mori Dream space for $n \geq 13$. Is there a similar ...
3
votes
1answer
131 views

Cohen-Macaulay rings in GIT

I'm starting to learn about Cohen-Macaulay rings in my commutative Algebra course, and my teacher mentioned its use in Geometric Invariant Theory. Having some basic knowledge on the subject and ...
2
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0answers
124 views

Projective and Quasiprojective quotients

Let $G$ be a finite group acting on a projective variety $X$. Then $G$ also acts on $X-X^G$, where $X^G$ is the fixed locus. The GIT quotient varieties $X/G$ and $(X-X^G)/G$ are projective and quasi-...
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149 views

Categorical quotients for quasi-affine varieties

Let $X$ be an affine variety and let $G$ be a reductive algebraic group acting on $X$. Let $U \subset X$ be a $G$-invariant open set. Under what hypothesis there exists a categorical quotient of $U$ ...
3
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1answer
151 views

What are the scalar conformal invariants of weight -3/2 in 3 dimensions?

I am looking for all the scalar conformal invariants (diffeomorphism-invariant polynomials $P[g]$ in the metric $g_{ij}$, its inverse $g^{ij}$ and its derivatives $g_{ij,klm\dots}$ such that $P[\...
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211 views

Quotients and Cohomology

Let $G=\mathbb Z_2$ and let $X \subset \mathbb P^5$ be a projective variety and the action of $G$ on $X$ is given by $1.(x_1,x_2,x_3,x_4,x_5,x_6)=-(x_6,x_5,x_3,x_4,x_2,x_1)$. I need to compute the ...
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292 views

Cohomology of the quotient

Let $X$ be a smooth projective variety on which a reductive group $G$ acts freely. When is it true that $H^*(X//G) \cong H^*(X)^G$ ? Here $X//G$ is the Mumford quotient with respect to a suitable $G$-...
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1answer
205 views

proj of an Algebra [closed]

Let $\mathbb Z_2= \langle\sigma\rangle$ act on $\mathbb C^6$ by $(x_1,x_2,x_3,x_4,x_5,x_6)=-(x_6,x_5,x_3,x_4,x_2,x_1)$. Then what is $\operatorname{Proj}\left(\left(\frac{\mathbb C[x_1,x_2,x_3,x_4,x_5,...
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407 views

Semistability of tensor products under automorphisms of tensored vector spaces

Let $A,B,C,D,E,F$ be vector spaces over a field. Let $x\in A \otimes B \otimes C$ and $y \in D \otimes E \otimes F$ be tensors that are semistable with respect to the natural actions of $\text{SL}(A) ...
14
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1answer
1k views

Why is Mumford's GIT-quotient so effective?

According to remark 6.14 in Shigeru Mukai's An introduction to invariants and moduli (unfortunately, the page is not available on Google Books, so I explain it below), the GIT-quotient of an affine ...
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94 views

Question about GIT: when is the map $\pi:X//_\theta G\rightarrow X//G$ birational?

I want to ask somebody who is more familiar with the theory of GIT quotients than I am, if there is a nice list of conditions on the action of a reductive group $G$ on an affine variety $X$ over ...
3
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0answers
137 views

Categorical quotient of open subsets of affine varieties

Let $X$ be a complex affine variety and $G$ be a complex reductive group acting on $X$. Let $X//G=\operatorname{Spec}\mathbb{C}[X]^G$ and $$\pi:X\to X//G$$ be the GIT quotient of $X$ by $G$. Suppose ...
4
votes
1answer
192 views

Supposed generalization of $X/(G \times H)\simeq (X/G)/H$ for GIT-quotients

I wonder whether it is true that the composition of two GIT-quotients is another GIT-quotient. It should be an analogue of a set-theoretic formula $X/(G \times H)\simeq (X/G)/H$ but with GIT-quotients ...
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0answers
261 views

Smooth quotients and separation of orbits

Consider a unipotent algebraic group $G$ over $\mathbb{C}$ acting polynomially on $\mathbb{C}^n$. Suppose that the quotient exists as an analytical geometric quotient, i.e., $\mathbb{C}^n/G$ is a ...
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2answers
334 views

Is this quotient of a threefold known? What are its singularities?

Assume $G$ is the Klein four group $G=\{1,\sigma_1,\sigma_2,\sigma_3\}$. Let $G$ act on $X=\mathbb{A}^2\times\mathbb{P}^1$ via: $$\sigma_1\cdot(x,y,[\lambda:\mu])=(-x,y,[\lambda:-\mu]) \text{ and } ...
4
votes
1answer
240 views

Vector bundles on quotient variety

Let an algebraic group $G$ act on a complex variety $X$ such that there is a good enough quotient $X/G$ (for example, $G$ acts on a vector space $V$ linearly and $X=V_{ss}$ is a variety of semi-stable ...
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165 views

Mumford's claim on the quasi-projectiveness of the coarse moduli of ppav over $\mathbb{Z}$

In paragraph 3 of chapter 7 of Mumford's Geometric Invariant Theory, the author proves that the coarse moduli space $A_{g,d,n}$ of abelian varieties of dimension $g$, with a degree $d^2$ polarization ...
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201 views

Can one construct the GIT quotient of a projective bundle?

Let $G=PGL(n)$ act on a smooth projective scheme $X$ over $\mathbb{C}$ with nontrivial finite stabilizers ($\cong \mathbb{Z}/2\mathbb{Z}$) only along a divisor $D\subset X$. Furthermore there a is a ...
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1answer
181 views

When is an almost geometric quotient flat?

All varieties here are over $\Bbb C$. Let $G$ be a reductive algebraic group acting algebraically on affine $n$-space $\Bbb A^n$. Let $R$ be the coordinate ring of $\Bbb A^n$. Assume that the natural ...
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0answers
106 views

GIT: For $x$ fixed, is $\{L:x \in X^s(L)\}$ open in $\text{Pic}^G(L)$?

Let $G$ be a complex reductive algebraic group acting on a complex variety $X$ (not necessarily projective) with $\text{Pic}^G(X)$ finite dimensional (for simplicity). For a fixed $x \in X$ define $$P^...
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1answer
250 views

Geometric invariant theory and normalizers of stabilizers

For simplicity, work over an algebraically closed field of characteristic $0$. Let $$\begin{aligned} X &= \text{a smooth projective variety,} \\ G &= \text{a reductive group acting linearly on ...
3
votes
1answer
161 views

How does grade projection act on homogeneous multivectors in geometric algebra?

I'm reading Clifford Algebra to Geometric Calculus by Hestenes, and struggling with an early result about reversion inside of a grade-projection operator. It is noted that $A_r$ and $B_s$ are ...
9
votes
1answer
348 views

Is the dimension of $V//G$ always the same as the dimension of $V^*//G$?

I would like to know whether there is an example of a reductive algebraic group $G$ (say, over the complex numbers $\mathbb{C}$) and a finite dimensional representation $V$ of $G$ such that dim$(V//G)$...
1
vote
1answer
237 views

Proj of some graded algebra

I was computing some GIT quotients and came up with the following question: to compute $\mathrm{Proj}(\mathbb C [f_1,f_2,f_3,f_4,f_5,f_6]/I)$ where $f_i$'s are homogeneous polynomials of same degree ...