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18
votes
4answers
2k views

When are GIT quotients projective?

Some background on GIT Suppose G is a reductive group acting on a scheme X. We often want to understand the quotient X/G. For example, X might be some parameter space (like the space of possible ...
14
votes
1answer
1k views

Why is the degree:rank ratio of a vector bundle called its “slope”?

Whenever one studies moduli spaces of vector bundles on curves, one of the first things to be introduced is the "slope" of a vector bundle, i.e., its degree:rank ratio. Is there a nice (preferably ...
12
votes
6answers
2k views

Why and how are moduli spaces of (semi)stable vector bundles well-behaved?

The slope of a vector bundle E is defined as mu(E) = deg(E)/rank(E). Then a vector bundle E is called semistable if mu(E') ≤ mu(E) for all proper sub-bundles E'. It is called stable if mu(E') < ...
12
votes
2answers
791 views

Is an affine “G-variety” with reductive stabilizers a toric variety?

Let $X=Spec(A)$ be a reduced normal affine scheme over an algebraically closed field $k$ of characteristic $0$, with an action of a connected reductive group $G$. Suppose $x\in X$ is a ...
11
votes
3answers
968 views

Can a coequalizer of schemes fail to be surjective?

Suppose $g,h:Z\to X$ are two morphisms of schemes. Then we say that $f:X\to Y$ is the coequalizer of $g$ and $h$ if the following condition holds: any morphism $t:X\to T$ such that $t\circ g=t\circ h$ ...
10
votes
1answer
575 views

When Are Quotients Complete Intersections?

Let $S_{n}$ denote the permutation group on $n$ letters and $G\subset S_{n}$ a transitive subgroup. The inclusion of $G$ in $S_{n}$ defines an action of $G$ on $\mathbb{C}^{n}$. By finding a ...
10
votes
1answer
629 views

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 $\sigma:G\times X\rightarrow X$. ...
10
votes
2answers
652 views

Rationality of GIT quotients

I recently worked through most of the proof of the rationality of the moduli of genus 3 curves, which seemed to have the following structure: Every nonhyperelliptic genus 3 curve is a smooth plane ...
10
votes
3answers
549 views

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 ...
10
votes
1answer
630 views

Are GIT's good categorical quotients just locally ringed space coequalizers?

Introduction: The definition of "good categorical quotient" in geometric invariant theory (given below) seems fairly ad hoc to me, except that it looks very similar to the coequalizer of the action in ...
9
votes
8answers
2k views

Resources on Invariant Theory

Hi, So my question is pretty much summed up by the summary - basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd ...
9
votes
2answers
625 views

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 ...
9
votes
1answer
999 views

Bad Categorical Quotients

Let G be an algebraic group acting on a scheme X. Then f: X --> Y is a categorical quotient if it is constant on G-orbits and any other G-invariant morphism factors through it in a unique fashion. ...
9
votes
2answers
646 views

Fukaya categories of hyperkahler reductions: general request for information

I'd really like to hear any references or information people have about the Fukaya categories of hyperkahler reductions of vector spaces (for more informations on the varieties, see Nick Proudfoot's ...
8
votes
3answers
1k views

Toric varieties as quotients of affine space

One way to define toric varieties is as quotients of affine $n-$space by the action of some torus. However, this is not strictly true as we need to throw away "bad points" which ruin this ...
8
votes
1answer
560 views

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 ...
8
votes
1answer
429 views

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 ...
7
votes
2answers
422 views

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 ...
7
votes
2answers
330 views

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, ...
7
votes
1answer
168 views

Invariants and stablizers for the $PGL(V)$ action on $End(V\otimes V)$

Let $K$ be a field of characteristic zero, and $V$ a finite dimensional vector space over $K$. Consider the action of the algebraic group $G:=PGL(V)$ on the vector space $W:=End_K(V^{\otimes 2})$ by ...
6
votes
3answers
649 views

Applications of non-reductive GIT

Geometric invariant theory works well when the algebraic group $G$ acting on a variety is reductive. There has been recent work by Doran and Kirwan here and here to find a canonical method of ...
6
votes
1answer
764 views

Hilbert-Mumford criterion and closedness

A version of the Hilbert-Mumford criterion states the following: Let $G$ be a linearly reductive group and $V$ a representation of $G$ over a field $k$ (alg. closed, char. zero). Suppose that $y \in ...
6
votes
1answer
741 views

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 ...
6
votes
1answer
193 views

GIT quotients and automorphisms

Let $X$ be a smooth projective variety. Then we have an exact sequence: $$0\mapsto Aut^{o}(X)\rightarrow Aut(X)\rightarrow H\mapsto 0$$ where $Aut^{o}(X)$ and $H$ are respectively the connected ...
5
votes
2answers
810 views

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 ...
5
votes
2answers
494 views

The canonical divisor of the Hilbert scheme $Hilb^n P^2$?

Hey everyone, I was wondering if anyone knows what the canonical divisor of the Hilbert scheme $Hilb^n P^2$ is --$Hilb^n P^2$ is the Hilbert scheme of degree-n zero dimensional subschemes of the ...
5
votes
2answers
221 views

Invariants of the maximal unipotent subgroup of GL(n) acting on the space of n by n matrices

Let $G=GL(n,\mathbb{C})$ and let $U\subset G$ be a maximal unipotent subgroup. (For example,assume that U is the set of upper triangular matrices with ones in the diagonal.) Now let ...
5
votes
1answer
347 views

If Spec(A) has a G-fixed point and a dense G-orbit, is Spec(A) a cone?

[Edited to include a dense orbit] Let $X=Spec(A)$ be a normal affine scheme over an algebraically closed field $k$, with an action of a linearly reductive group $G$. Suppose $x\in X$ is a ...
5
votes
0answers
324 views

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 ...
4
votes
3answers
916 views

Why can we define the moment map in this way (i.e. why is this form exact)?

Given a symplectic manifold $(X, \omega)$ and a group $G$ acting on $X$ preserving the symplectic form, we define the moment map $\mu : X \to \mathfrak{g}^*$ so that $$ \langle d\mu(v), \xi\rangle = ...
4
votes
1answer
1k views

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 ...
4
votes
4answers
1k views

Near Trivial Quiver Varieties

So, today I started learning the definition of a quiver variety, and wanted to make sure I'm understanding things right, so first, my setup: I've been looking at the simplest case that didn't look ...
4
votes
2answers
383 views

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 ...
4
votes
2answers
745 views

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 ...
4
votes
1answer
161 views

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, ...
4
votes
2answers
291 views

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 ...
4
votes
1answer
300 views

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 ...
4
votes
2answers
1k views

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 ...
4
votes
1answer
187 views

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 ...
4
votes
0answers
132 views

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 ...
4
votes
0answers
125 views

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 ...
4
votes
0answers
105 views

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 ...
4
votes
0answers
315 views

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 ...
4
votes
0answers
160 views

Components of variety of subalgebras

This question is motivated by the question Subalgebras of matrices and its answer by Mariano. We consider $X_{n,d}$, the variety of $d$-dimensional subalgebras not necessarily with 1 (with 1 makes ...
3
votes
2answers
374 views

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 ...
3
votes
2answers
316 views

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 $E_7$.
3
votes
1answer
245 views

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 ...
3
votes
2answers
238 views

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 ...
3
votes
1answer
281 views

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 ...
3
votes
2answers
201 views

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 ...