All Questions
Tagged with geometric-invariant-theory ag.algebraic-geometry
81 questions with no upvoted or accepted answers
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
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
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
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
5
votes
0
answers
351
views
What representation theoretic properties does the semi-invariant ring tell us?
I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)?
I have been studying about semi-...
- 839
5
votes
0
answers
245
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 ...
- 3,318
5
votes
0
answers
347
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-...
- 558
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
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
5
votes
0
answers
1k
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 ...
- 51
5
votes
0
answers
219
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 ...
- 101
5
votes
0
answers
165
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 ...
- 101
4
votes
0
answers
310
views
GIT quotient of a reductive Lie algebra by the maximal torus
Let $G$ be a connected complex reductive group with Lie algebra $\mathfrak{g}$. One knows a lot about the GIT quotient $\mathfrak{g}/\!/G$: the invariant ring is a free polynomial algebra on $\mathrm{...
- 2,751
4
votes
0
answers
227
views
Elementary proof Hilbert-Mumford stability criterion for $\operatorname{GL}_n(\mathbb{C})$
In An elementary proof of the Hilbert-Mumford criterion, B. Sury gives an elementary proof of the Hilbert-Mumford semi-stability criterion for $G = \operatorname{GL}_n(\mathbb{C})$ (and $G = \...
- 7,320
4
votes
0
answers
98
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 ...
- 599
4
votes
0
answers
113
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 ...
- 1,383
4
votes
0
answers
271
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 ...
- 141
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
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 $\...
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
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
4
votes
0
answers
167
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 ...
- 12.4k
3
votes
0
answers
126
views
Parametrization of indecomposable modules via quiver varieties
Let $k$ be an algebraically closed field, $Q$ a quiver without oriented cycles and $m^\alpha (Q)$ the variety of quiver representations with dimension vector $\alpha$. There is a canonical algebraic ...
- 696
3
votes
0
answers
267
views
Does the orbit in geometric invariant theory have natural scheme structure
Let $X$ be a scheme locally of finite type over a sufficiently "nice" base scheme $S$ (nice in sense of reasonable "finiteness conditions", for sake of simplicity let's start as ...
- 5,966
3
votes
0
answers
345
views
On Noether's Problem
Noether's problem is a famous problem in invariant theory, introduced in the 1910's by Emmy Noether in relation to the inverse Galois problem. It is as follows:
Noether's Problem: Let $F=k(x_1,\dotsc,...
- 3,318
3
votes
0
answers
287
views
When a stack quotient coincides with GIT quotient?
Let $G$ be a reductive group over $\mathbb{C}$, and $H=H_r\ltimes H_u$ be a subgroup of $G$. Here, $H_u$ is unipotent and $H_r$ is reductive.
Question: Is it true that when $G/H$ is open in its affine ...
- 31
3
votes
0
answers
164
views
Class of finite quotient affine space in Grothendieck ring of varieties
Let $G$ be a finite group acting linearly on affine space $\mathbb{A}^n_k$ over $k = \mathbb{C}$. Since the action is linear, it can be extended to an action of $\operatorname{GL}_n(\mathbb{C})$, ...
- 197
3
votes
0
answers
175
views
Nef cone of a GIT quotient
I want to know how to calculate nef cone of a GIT quotient. In particular let $X$ be a projective variety and $L$ be an ample line bundle on $X$ and $G$ be a reductive algebraic group and chosen a $G$ ...
- 61
3
votes
0
answers
147
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 $k$ (...
- 14.2k
3
votes
0
answers
149
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 ...
- 5,039
3
votes
0
answers
235
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 ...
- 4,600
3
votes
0
answers
140
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 $X$. ...
- 31
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
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
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
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
2
votes
0
answers
182
views
GIT quotient and orbifolds
Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...
- 2,751
2
votes
0
answers
137
views
Is the GIT quotient of a finite map of varieties again a finite map?
Let $K$ be an algebraically closed field of characteristic $0$, let $X/K$ and $Y/K$ be quasi-projective varieties, and let $f:X\to Y$ be a morphism. Let $G/K$ be a reductive group that acts stably on $...
- 47.4k
2
votes
0
answers
188
views
Help with Macaulay2 computation of invariant ring
Consider the algebraic group $G:=\operatorname{SL}_{2}\times\operatorname{SL}_{2}$ acting on $V:=\operatorname{Mat}_{2\times 2}\oplus\operatorname{Mat}_{2\times 2}$ via the action $(A,B)\,\cdot\,(X,Y)=...
- 839
2
votes
0
answers
174
views
How are tangent spaces related via geometric quotient?
Let $G$ be a linearly reductive group acting regularly on an irreducible affine variety $X$ (over an algebraically closed field of characteristic zero). Suppose there's a $G$-stable open subvariety $U$...
- 839
2
votes
0
answers
98
views
What is the natural linearization on differentials?
Let $\Bbbk$ be a field. Let $G$ be an affine algebraic group over $\Bbbk$. Let $X$ be a scheme over $\Bbbk$. Let $G$ act on $X$ with the action morphism $\sigma:G\times X\to X$. There are two ...
- 930
2
votes
0
answers
184
views
Finding étale slices
I'm trying to understand Luna's étale slice theorem by computing some examples. The theorem is usually phrased as an existence result. I wondered if there was a natural way to figure out the slice at ...
- 185
2
votes
0
answers
290
views
GIT-quotients of projective-over-affine varieties
Given an action of a reductive group on a projective-over-affine variety, what are the conditions for its GIT-quotient to be again a projective-over-affine?
There is a very nice set of slides about ...
- 33
2
votes
0
answers
186
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{\...
- 197
2
votes
0
answers
306
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. ( ...
- 815
2
votes
0
answers
104
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 ...
- 3,031
2
votes
0
answers
319
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