Questions tagged [geometric-invariant-theory]

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

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3
votes
0answers
75 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 $k$ (...
5
votes
1answer
161 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? ...
2
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1answer
106 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 \...
4
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1answer
176 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$...
4
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1answer
128 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 ...
3
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1answer
112 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}(...
2
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1answer
195 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 ...
0
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1answer
156 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 ...
4
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0answers
60 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 ...
3
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1answer
235 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 $X$ ...
0
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0answers
102 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 ...
2
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0answers
80 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{\...
4
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0answers
81 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 ...
7
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1answer
413 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 ...
7
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1answer
224 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 \...
5
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0answers
148 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 ...
1
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1answer
164 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 ...
12
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1answer
315 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. ...
3
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0answers
76 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 ...
1
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0answers
111 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 ...
4
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0answers
159 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 ...
2
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0answers
136 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. ( ...
3
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1answer
134 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:...
3
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0answers
142 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 ...
5
votes
1answer
300 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 \...
0
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2answers
330 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 ...
1
vote
1answer
120 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 ...
5
votes
1answer
724 views

Preparation for GIT (Geometric Invariant Theory)

I am trying to read Mumford's Geometric Invariant Theory, however, I find my knowledge in algebraic geometry is inadequate. My knowledge is at the level of Hartshorne's Algebraic Geometry. Mumford ...
3
votes
1answer
171 views

Quotient of a Fano variety by a torus

We work over an algebraically closed field of characteristic zero. Let $X$ be a Fano variety, and $T\cong \mathbb{G}_m^r$ a torus acting faithfully on $X$. I think we can canonically linearize the ...
4
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0answers
155 views

Invariants of linear endomorphisms of tensor products

Let $V$ and $W$ be two finite dimensional vector spaces over an algebraically closed field $K$ of characteristic zero. Consider the coordinate ring $K[\mathrm{End}(V\otimes W)]$ of the affine space ...
3
votes
0answers
160 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
votes
0answers
73 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 $−2πi\Omega$ ...
2
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0answers
91 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
votes
1answer
68 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 ...
2
votes
1answer
140 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
votes
0answers
236 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-...
3
votes
0answers
113 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 $X$. ...
4
votes
0answers
113 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
241 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 ...
1
vote
1answer
224 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 $...
4
votes
1answer
325 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 ...
10
votes
0answers
197 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 $V$. ...
6
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0answers
131 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{...
1
vote
1answer
133 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 $Y$ ...
1
vote
0answers
27 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 ...
1
vote
0answers
128 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
votes
2answers
516 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
votes
1answer
144 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
votes
1answer
617 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
votes
1answer
437 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 ...