The geometric-invariant-theor tag has no wiki summary.

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### Hilbert point and Hilbert stability

For $X\in \mathbb{P}^N$ a closed subscheme, one can consider the m-th Hilbert point
$$
[X]_m=[\bigwedge^{h^0(X, \mathcal{O}(m))}H^0(\mathbb{P}^N, \mathcal{O}(m))\to \bigwedge^{h^0(X, ...

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### Why we study Geometric invariant theory?

I am trying to learn Geometric invariant theory like it was introduced by Mumford. But I do not have a strong motivation and so I want to know the reason of studying Geometric invariant theory. I just ...

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### Semistability in GIT

If I understand correctly, in geometric invariant theory, polystable points can be defined as those which have a closed orbit. Is it true that semistable points can be characterized as those whose ...

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### When a Spherical variety is $K$-stable

Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical.
My question is When a Spherical variety is $K$-stable? Is ...

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### stability notion of nets of quadrics

A net of quadrics in $\mathbb{P}^n$ is a plane in $\mathbb{P}^N$, where $N=\frac{n(n+3)}{2}$. So the space of net of quadrics is the Grassmannian $Gr(3,N+1)$. The group $SL_{n+1}(\mathbb{C})$ acts on ...

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### (semi)stability of a net of quadrics and a certain criterion

Let $V:=H^0(\mathbb{P}^4,\mathcal{O}(1))$, $W:=H^0(\mathbb{P}^4,\mathcal{O}(2))$ and $Q_i(i=1,2,3)$ be quadric hypersurfaces in $\mathbb{P}^4$.
To a net of quadrics $\Lambda=(Q_1,Q_2,Q_3)$, we ...

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### Research of a reference about $G$-linearizations of line bundles on quasi-projective schemes

I am looking for some references for the following statement:
Let $G$ be a linearly reductive algebraic group acting on a quasi-projective scheme $X$, over an algebraically closed field $K$. Let $L$ ...

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### Quotients by the additive group $\mathbb G_a$

Geometric invariant theory doesn't work so well for non-reductive groups, since invariant rings are not generally finitely generated. However, in many cases the action of a non-reductive group has a ...

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### How does one show that slope stability of a vector bundle is an open condition with respect to the polarisation?

I would like a source for the following result, which I expect to be true (probably well known):
Let $X$ be a complex projective variety, $L$ an ample line bundle and $E$ a slope stable vector bundle ...

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### Smoothness of fix point components of finite group action on smooth variety

Let $X$ be a smooth complex algebraic variety, and $\varphi: \Gamma\curvearrowright X$ an action (by automorphisms) of a finite group $\Gamma$ on $X$.
Can we say that each irreducible component of ...

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### A question about Marsden-Weinstein reduction theory

Let $G$ be a compact Lie group and $\frak g$ be its Lie algebra. Then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J \colon M\to \frak g^*$ be its moment map then the ...

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### A quotient stack question

Let $X$ be a proper Deligne-Mumford stack, whose normalization, $X'$, is a global quotient stack (that is, a stack of the form [W/GL_n],where W is an algebraic space) with a projective scheme as a ...

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### Understanding the definition of the quotient stack $[X/G]$

I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's Algebraic Stacks and Moduli of Vector Bundles.
Explicitly, let $G$ be an affine smooth group ...

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### Why people usually consider reductive groups in GIT?

Where do people essentially use the reductive groups in the theory of GIT? Or how does reductive groups simplify the constructions in GIT?
I found that the property of completely reducible of ...

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### Quotient of a rational variety by a finite group

Let $X$ be a rational variety and let $G$ be a finite group acting on $X$. Let us consider the diagonal action of $G$ over the product $X^{h} = X\times...\times X$,
$$G\times(X\times...\times ...

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### Finding relations between invariant polynomials

Suppose I have an action of a linear reductive group ($GL(2,\mathbb{C})^2$ in this case) on a complex vector space (of dimension $16$) and I want to compute explicitly the ring of invariants of this ...

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### when does one want to use the Reynolds operator in GIT?

The role of Reynolds operator in GIT has always been a little mystery to me. Actually I see that in some proofs it gets used in an efficient way, but what I cannot grasp is the general philosophy. I ...

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### Scaling-Invariant Orbits of Semisimple Group Representations

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Note that if $V$ is the adjoint representation of $G$, then ...

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### When does a group action on a k-algebra induce an algebraic action on the spectrum?

This question arose from my last question, which I considered answered - from the comments, however, it is obvious that the answer is only complete in characteristic zero, and I am trying to ...

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### Equivariant fibre product

Let $G$ be an algebraic group. Let $X$ and $Y$ be $S$-schemes such that $X$, $Y$ and $S$ are $G$-schemes and the structural morphisms are equivariant. My question is: Can the fiber product ...

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### Are orbits of an affine algebraic monoid affine?

Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let ...

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### non-flat GIT quotient

Let $G=PGL(N)$ acting on a scheme $X$ over a field $k$ and $L$ be a $G$-linearized invertible sheaf. Let $X^{ss}(L)$ be the semistable locus. We know that a uniform categorical quotient ...

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### Partial (or complete) flag varieties as GIT quotients of affine spaces

I am looking for presentations of partial or complete flag varieties as GIT quotients of affine varieties spaces. That is, for a choice of of dimensions $0=d_1<d_2<\dots<d_k = n$, I would ...

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### Coarse moduli spaces of quotient stacks

Suppose you have a separated Deligne Mumford quotient stack $[V/G]$ over a field of characteristic $0$, where $V$ is a quasiprojective variety and $G$ is an algebraic group that does not necessarily ...

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### Normalization of quotient stacks

Suppose you have a Deligne Mumford stack which is a quotient $[X/G]$ of a scheme $X$ by an algebraic group $G$ .
What is the normalization of that? Is it true that its normalization is a quotient ...

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### question about relative stable maps

Let $C$ be a connected smooth curve, $0\in C$ a closed point and $W\rightarrow C$ a family of projective schemes. Assume that the fibers $W_t$ of $W$ are smooth for all $t\neq 0$ and that $W_0=Y_1\cup ...

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### When does a G-invariant one to one map between two closed algebraic G-set descend to a one to one map on the G.I.T quotient ?

I do not know much about Geometric Invariant Theory. My question is the following:
Let $X$ and $Y$ be two complex affine or projective varieties. Let $G$ be a reductive group which acts on both $X$ ...

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### blow up of segre primal and $\mathcal{M}_{0,6}$

The segre cubic primal $X\subset P^4$ is the GIT quotient of 6 points on $P^1$. Let $M_{0,6}$ the DM compactification of the moduli of 6-pointed rational curves. The Segre primal $X$ is a cubic 3-fold ...

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### degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$

How can we find the degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ , $dimV=8$ by the model in the Lie algebra $E_7$.

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### Quotient of affine space by cyclic permutation

The quotient of the affine space $\mathbb{A}^n$ by the symmetric group $Sym_n$ is again an affine space of the same dimension, and invariants are given by elementary symmetric polynomials.
What ...

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### Algebraic closure and GIT

Does one need to work over an algebraic closed field in ordre to construct GIT quotients à la Mumford?
If yes, why?

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### What does this particular geometric quotient locally look like?

Let $k$ be a field and consider the algebraic group $GL_n$ over $Spec(k)$. It has as a closed (but not normal) algebraic subgroup the group $M$ of monomial matrices, i.e. matrices having exactly one ...

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### Intersection theory for $G$-varieties - an action on the chow ring?

Let $G$ be a reductive algebraic group. Let $X$ be a $G$-variety and consider any closed subvariety $Z$ of $X$. Since any $g\in G$ acts as an automorphism, we know that $g.Z$ is again a closed ...

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### A line bundle that does not admit a G-linearisation

I have been thinking about quotients lately and pondered the following:
Let $G$ be a connected linear algebraic group and $X$ a $G$-variety acting via the morphism $\sigma:G\times X\rightarrow X$. ...

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### Algorithms in Invariant Theory

Let $V$ be a polynomial representation of the general linear group $\Gamma:=\DeclareMathOperator{\Gl}{Gl}\Gl_n(\newcommand{\C}{\mathbb C}\C)$.
In chapter 4.6 of his book "Algorithms in Invariant ...

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### When is an orbit spherical?

I asked the following question over at math.stackexchange, but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here:
Let's assume we have an affine, reductive, ...

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### Lift of a morphism between geometric quotients

Let $S$ be a scheme.
Definition. Let $X$ be an $S$-scheme and $G$ a smooth affine group $S$-scheme acting on $X.$ An $S$-scheme $Y$ is a geometric quotient of $X$ by $G$ if there exists a morphism ...

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### About the strength of representation-theoretic obstructions for orbit closure problems

Let $G$ be a reductive, affine, algebraic group over $\newcommand{\C}{\mathbb C}\C$. Let $X$ be a $G$-variety. For $x\in X$, we write
$$G_x:=\{ g\in G\mid g.x=x\}$$
for its stabilizer and for any ...

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### Stability conditions for coherent sheaves and GIT

I am learning stability conditions for derived categories of coherent sheaves, following Bridgeland, and coming from a vector bundles background. $\mu$-stability for vector bundles has a clear GIT ...

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### invariants of plane quartics

Does anybody know a good reference where the invariants for plane quartic curves are developed?

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### Action of k* on a variety induces grading?

Let $V$ be a $\Bbbk$-variety such that $\Bbbk^\times$ (as an algebraic group) acts algebraically on $V$. Given any $f\in\Bbbk[V]$, let us call $f$ homogeneous of degree $d$ if for all $v\in V$ and all ...

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### affinization of T^*CP^n

Is there an elementary description of the affinization of the algebraic cotangent bundle of $CP^n$? I know that it can be described as some sort coadjoint orbits, but I am interested in a translation ...

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### When the affine quotient is faithfully flat?

It may be easy for the expert.
Consider the map from $n$ by $m$ matrices (over $\mathbb{C}$ )to the $n$ by $n$ symmetric matrices $\phi\colon A\mapsto A A^T$.
My question is when this map is ...

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### Quotients of group actions on varieties

Let $Y$ be an affine algebraic variety over $\mathbb{C}$ and let $X$ be its closed subvariety. Let $G$ be a reductive group acting on $Y$ and let $H$ be a reductive subgroup of $G$ preserving $X$ such ...

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### Level n-structure as defined by Mumford in GIT

In Mumford's GIT, the definition of level $n$ structure ($n \geq 2)$ is $2g$ sections $\{\sigma_1, \dots, \sigma_{2g}\} : S \rightarrow A$ such that two conditions hold: (i) For geometric points the ...

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### Hilbert polynomial of an abelian scheme

This is coming out of Mumford's GIT, section 7.2, page 131.
$A/S$ an abelian scheme of dimension $g$ with polarization $\bar{\omega}$ of degree $d^2$. Then $\pi_*(L^\Delta(\bar{\omega})^3)$ is ...

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### Verifying claims in the proof of the Rigidity Lemma (Mumford, GIT)

In Chapter 6 of Mumford's Geometric invariant theory, during the proof of the rigidity lemma, there are two statements I'm not sure how to verify. The general setup is:
$p : X \rightarrow S$ is ...

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### References to SGA 8 and descent theory

In Geometric Invariant Theory, by Mumford, Fogarty, and Kirwan, if there is a mention of descent theory, it almost always comes along with a reference to SGA 8, Theorem 5.2 (see the end of the proof ...

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### Configuration space of flags

Let $U\subset \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be the Zariksi open set of ordered quadruple of distinct points in the projective line. The quotient of $U$ by the ...

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### A simple problem on commutative algebra related to G.I.T

Let $G$ be a geometrically reductive algebraic group over an algebraically closed field $k$. Let $X$ be an affine variety over $k$ on which $G$ acts regularly. Then $G$ acts on the coordinate ring $A$ ...