All Questions
Tagged with geometric-invariant-theory ag.algebraic-geometry
195 questions
2
votes
1
answer
721
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 ...
- 185
3
votes
1
answer
760
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 ...
- 297
3
votes
0
answers
325
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^...
- 113
5
votes
0
answers
146
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 ...
- 815
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
1
vote
1
answer
189
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 ...
- 507
2
votes
0
answers
318
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 ...
- 501
2
votes
0
answers
172
views
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-...
- 213
4
votes
0
answers
406
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 $U$ ...
- 63
-1
votes
1
answer
230
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,...
- 95
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
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
2
votes
0
answers
107
views
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 ...
- 790
3
votes
0
answers
213
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 ...
- 779
5
votes
1
answer
322
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 ...
- 1,980
5
votes
0
answers
278
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 ...
- 189
3
votes
2
answers
371
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:
$$\sigma_1\cdot(x,y,[\lambda:\mu])=(-x,y,[\lambda:-\mu]) \text{ and } ...
- 1,025
4
votes
1
answer
628
views
Vector bundles on quotient variety
Let an algebraic group $G$ act on a complex variety $X$ such that there is a good enough quotient $X/G$ (for example, $G$ acts on a vector space $V$ linearly and $X=V_{ss}$ is a variety of semi-stable ...
- 1,980
1
vote
0
answers
189
views
Mumford's claim on the quasi-projectiveness of the coarse moduli of ppav over $\mathbb{Z}$
In paragraph 3 of chapter 7 of Mumford's Geometric Invariant Theory, the author proves that the coarse moduli space $A_{g,d,n}$ of abelian varieties of dimension $g$, with a degree $d^2$ polarization ...
user85435
3
votes
0
answers
269
views
Can one construct the GIT quotient of a projective bundle?
Let $G=PGL(n)$ act on a smooth projective scheme $X$ over $\mathbb{C}$ with nontrivial finite stabilizers ($\cong \mathbb{Z}/2\mathbb{Z}$) only along a divisor $D\subset X$. Furthermore there a is a ...
- 1,025
3
votes
1
answer
619
views
When is an almost geometric quotient flat?
All varieties here are over $\Bbb C$. Let $G$ be a reductive algebraic group acting algebraically on affine $n$-space $\Bbb A^n$. Let $R$ be the coordinate ring of $\Bbb A^n$. Assume that the natural ...
- 3,079
1
vote
0
answers
111
views
GIT: For $x$ fixed, is $\{L:x \in X^s(L)\}$ open in $\text{Pic}^G(L)$?
Let $G$ be a complex reductive algebraic group acting on a complex variety $X$ (not necessarily projective) with $\text{Pic}^G(X)$ finite dimensional (for simplicity). For a fixed $x \in X$ define
$$P^...
- 691
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
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
1
vote
1
answer
306
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 ...
- 125
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
5
votes
1
answer
438
views
A criterion for orbits of complex reductive group to be closed
I am having some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following:
Let $G=K_{\Bbb C}$ be a ...
- 779
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
4
votes
0
answers
169
views
Quotients of quasi affine varieties and extension of scalars
I have some questions about GIT quotients and extensions of scalars of categorical quotients:
1) Let $X$
be a complex algebraic quasi-affine variety, $G$
an algebraic reductive group over $\...
0
votes
1
answer
167
views
Smoothness and quotient
Suppose we have a smooth Mumford's quotient $Q//PGL_k(m)$ where $Q$ is a quasi-projective variety and $k$ is an algebraically closed field of positive characteristic. Is it true that $Q$ is also ...
- 155
3
votes
0
answers
364
views
Choosing a group action to do GIT of hypersurfaces
When studying GIT stability of hypersurfaces $d$ of $\mathbb P^n$ we look at the Hilbert Scheme $H=\mathbb P^N$ parametrizing homogeneous polynomials $f_d(x_0,\ldots,x_n)$ of degree $d$. There is ...
- 31
3
votes
1
answer
313
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$$
where $Aut^{o}(X)$ and $H$ are respectively the connected ...
user77919
6
votes
2
answers
701
views
Invariants of the maximal unipotent subgroup of GL(n) acting on the space of n by n matrices
Let $G=GL(n,\mathbb{C})$ and let $U\subset G$ be a maximal unipotent subgroup. (For example,assume that U is the set of upper triangular matrices with ones in the diagonal.) Now let $X=M_{n}(\mathbb{C}...
- 375
5
votes
1
answer
383
views
Hilbert point and Hilbert stability
For $X\in \mathbb{P}^N$ a closed subscheme, one can consider the m-th Hilbert point
$$
[X]_m=[\bigwedge^{h^0(X, \mathcal{O}(m))}H^0(\mathbb{P}^N, \mathcal{O}(m))\to \bigwedge^{h^0(X, \mathcal{O}(m))}H^...
- 83
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
2
votes
2
answers
2k
views
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 ...
- 1,347
1
vote
0
answers
187
views
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 $L$ ...
- 401
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
3
votes
1
answer
670
views
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 ...
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
4
votes
0
answers
520
views
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 ...
- 73
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
5
votes
2
answers
867
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 X)\...
- 8,998
1
vote
1
answer
482
views
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 ...
- 3,779
4
votes
0
answers
140
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 ...
- 4,920
5
votes
1
answer
608
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 ...
- 4,902
1
vote
1
answer
450
views
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 $P:=X\...
- 4,902
1
vote
2
answers
296
views
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 ...
- 4,902
1
vote
0
answers
266
views
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 $\phi:X^{ss}(L)...
- 61
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