All Questions
Tagged with geometric-invariant-theory ag.algebraic-geometry
195 questions
25
votes
4
answers
4k
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 ...
- 24.1k
24
votes
2
answers
4k
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 ...
- 7,338
20
votes
10
answers
7k
views
Resources on invariant theory
What are resources on invariant theory? 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 prefer online / freeish ...
19
votes
6
answers
4k
views
Why and how are moduli spaces of (semi)stable vector bundles well-behaved?
The slope of a vector bundle $E$ is defined as $\mu(E) = \deg(E)/\mathrm{rank}(E)$. Then a vector bundle $E$ is called semistable if $\mu(E') \leqslant \mu(E)$ for all proper sub-bundles $E'$. It is ...
- 21k
19
votes
3
answers
2k
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 $t\circ g=t\circ h$ ...
- 24.1k
19
votes
1
answer
2k
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 where the action is the morphism $\sigma:G\times X\...
- 1,201
18
votes
1
answer
3k
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 ...
- 1,980
17
votes
3
answers
2k
views
Variety of commuting matrices
Let $G=\operatorname{GL}(n,\mathbb{C})$ and $\mathfrak{g}=\operatorname{Mat}(n,\mathbb{C})$ and let us consider the two varieties $X,Y$ defined as $$X=\{(x,y) \in G \times G \ | \ xy=yx\} $$ and $$Y=\{...
- 1,061
16
votes
2
answers
2k
views
Bad Categorical Quotients
Let $G$ be an algebraic group acting on a scheme $X$. Then $f: X \to Y$ is called a categorical quotient if it is constant on $G$-orbits and every $X \to Z$ constant on $G$-orbits factors through it ...
- 2,806
13
votes
2
answers
1k
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 ...
- 16k
13
votes
1
answer
1k
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 ...
- 11.3k
12
votes
1
answer
502
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. ...
- 297
12
votes
2
answers
1k
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 $G$-...
- 24.1k
11
votes
2
answers
2k
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 ...
- 142
11
votes
1
answer
918
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 ...
- 2,097
11
votes
2
answers
684
views
Invariants of $\mathrm{GL}_n$ representations
$\DeclareMathOperator\GL{GL}$Let $V=\mathbb C^n$ be the natural representation of $\GL_n(\mathbb C)$ and let $W=\operatorname{Sym}^2(V)$ be the symmetric square representation. Let $W^k$ denote the ...
- 673
11
votes
3
answers
2k
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 construction....
- 21.4k
11
votes
1
answer
938
views
An easy textbook for geometric invariant theory and moduli space which makes use of scheme theory
I would like to study geometric invariant theory and moduli theory.
It seems that a standard textbook for these fields is "Geometric Invariant Theory" written by D.Mumford, J.Fogarty and F....
- 197
11
votes
3
answers
961
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 ...
- 262
11
votes
0
answers
451
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) ...
- 149k
10
votes
2
answers
994
views
Character variety of the free group
A classical result of Fricke--Klein--Vogt from the late 1800s implies that the character variety $\chi_\mathbb{C}$ associated to the free group $F_2$ and the algebraic group $\mathrm{SL}_2(\mathbb{C})$...
- 2,751
10
votes
1
answer
1k
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 ...
- 3,472
10
votes
2
answers
1k
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 ...
- 44.7k
9
votes
3
answers
2k
views
Quotient of smooth algebraic variety by proper free action of algebraic group
Let $G$ be algebraic group acting on a smooth algebraic variety $M$. Assume the action is proper and free. Is the orbit space $M/G$ an algebraic variety? If so, could someone point to a reference? If ...
- 103
9
votes
1
answer
1k
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 ...
- 4,902
9
votes
1
answer
416
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)$...
- 543
9
votes
1
answer
346
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 ...
- 213
8
votes
2
answers
1k
views
Affine GIT is an open map?
Let $k$ be a field, $X= \text{Spec}\,A$ be an affine scheme, with $A$ a finitely generated $k$-algebra. $G=\text{Spec}\,R$ is a linearly reductive group acting rationally on A, i.e. every element of $...
- 355
8
votes
2
answers
497
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, ...
- 4,902
8
votes
1
answer
379
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 ...
- 47.4k
8
votes
1
answer
1k
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 ...
- 419
8
votes
1
answer
493
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 ...
- 4,902
8
votes
0
answers
235
views
Stability of nodal hypersurfaces
We denote by $\Pi_{n,d}$ the space of homogeneous polynomials of degree $d$ in $n+1$ variables $x_0,\ldots,x_n$, i.e. $\Pi_{n,d}=\Gamma(\mathbb{P}^n(\mathbb{C}),\mathcal{O}(d))$. The group $G=SL(n+1)$ ...
7
votes
1
answer
3k
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 ...
- 87
7
votes
2
answers
772
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 ...
- 71
7
votes
3
answers
1k
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 ...
- 508
7
votes
1
answer
456
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 ...
- 10.4k
7
votes
1
answer
397
views
Is there a Chevalley map for spherical varieties?
If $G$ is a reductive group, $T$ a maximal torus and $W$ its Weyl group the Chevalley restriction theorem (in its "multiplicative" version) gives an isomorphism between the GIT quotient of $...
- 553
7
votes
1
answer
305
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 \...
- 3,318
7
votes
1
answer
718
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 ...
- 1,383
7
votes
1
answer
835
views
Intuition for Luna's Étale Slice Theorem
I am trying to understand the intuition for Luna's Étale Slice Theorem in the affine setting over $\mathbb{C}$.
Here is the setup. Let $X$ be an affine algebraic variety and $G$ a reductive group ...
- 83
7
votes
0
answers
466
views
Kähler quotients of affine varieties and GIT
Let $X\subseteq \Bbb C^n$ be a smooth affine variety and $G=K_{\Bbb C}$ a complex reductive group acting linearly on $\Bbb C^n$ preserving $X$ (where $K$ is a maximal compact subgroup of $G$). Suppose ...
- 779
6
votes
2
answers
701
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 $X=M_{n}(\mathbb{C}...
- 375
6
votes
1
answer
2k
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 ...
- 63
6
votes
1
answer
2k
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 ...
- 23.4k
6
votes
3
answers
2k
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 flat,...
- 419
6
votes
0
answers
679
views
Learning about moduli spaces of sheaves
I am a Ph.D. student and starting a side project with a fellow student on Moduli spaces. Our plan was to start with the book on Invariants and Moduli by Mukai (starting from chapter 5) and use the ...
- 345
6
votes
0
answers
591
views
Affine GIT quotients and the excursion algebra in Fargues–Scholze
Some background:
Let us fix a non-archimedean local field $E$ with residue characteristic $p$, and let $G$ be some connected reductive group over $E$. In [FS, §VIII.1.1] the authors define a moduli ...
- 787
6
votes
0
answers
141
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{...
- 17.4k
6
votes
0
answers
544
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 ...
- 3,779