# Questions tagged [geometric-invariant-theory]

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

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### 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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**0**answers

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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### 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**

votes

**1**answer

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In unpublished notes by Yi Hu (which appear to be no longer online), I found the following: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}=...

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**1**answer

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})^{...

**5**

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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

**1**answer

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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**1**answer

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 $...

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**1**answer

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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**1**answer

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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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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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 $\...

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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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**1**answer

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^{...

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**1**answer

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 ...

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**1**answer

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 ...

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**1**answer

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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**1**answer

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 ...

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votes

**1**answer

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 ...

**2**

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**0**answers

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 ...

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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 ...

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### 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**

votes

**1**answer

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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### 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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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) ...

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votes

**1**answer

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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### 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 ...

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votes

**0**answers

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 ...

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votes

**1**answer

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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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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votes

**2**answers

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 } ...

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votes

**1**answer

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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**1**answer

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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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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**1**answer

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

**1**answer

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

**1**answer

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**

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**1**answer

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 ...