All Questions
Tagged with geometric-invariant-theory ag.algebraic-geometry
195 questions
5
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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 ...
- 51
5
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0
answers
219
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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 ...
- 101
5
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0
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165
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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 ...
- 101
2
votes
2
answers
530
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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 $X$ ...
- 481
3
votes
2
answers
334
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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 ...
- 3,779
4
votes
2
answers
757
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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 ...
- 7,722
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1
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324
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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?
- 3,779
1
vote
1
answer
282
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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 ...
- 319
3
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2
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367
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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 ...
- 4,902
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1
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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 where the action is the morphism $\sigma:G\times X\...
- 1,201
11
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3
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961
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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 ...
- 262
8
votes
2
answers
497
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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, ...
- 4,902
4
votes
1
answer
369
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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 $\...
user17778
8
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1
answer
493
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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 ...
- 4,902
6
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544
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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 ...
- 3,779
1
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1
answer
219
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invariants of plane quartics
Does anybody know a good reference where the invariants for plane quartic curves are developed?
- 3,779
9
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1
answer
1k
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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 ...
- 4,902
5
votes
2
answers
426
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Affinization of $T^*\mathbb{C}P^n$
Is there an elementary description of the affinization of the algebraic cotangent bundle of $\mathbb CP^n$? I know that it can be described as some sort coadjoint orbits, but I am interested in a ...
3
votes
1
answer
310
views
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 ...
- 157
1
vote
1
answer
406
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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 ...
- 2,390
2
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0
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218
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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 ...
- 419
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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 ...
- 419
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3
answers
2k
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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 flat,...
- 419
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1
answer
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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 ...
- 419
1
vote
2
answers
576
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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 ...
- 1,804
2
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0
answers
363
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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 $A$ ...
- 1,804
24
votes
2
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4k
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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 ...
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9
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3
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2k
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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
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2
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1k
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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
5
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1
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470
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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 $G$-...
- 24.1k
13
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1
answer
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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
3
votes
1
answer
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geometric quotient
Let $S$ be a base scheme. Let $X$ be a scheme over $S$ and let $G$ be a group scheme over $S$ acting on $X$ via $\sigma: G \times_S X \to X$. Suppose that we have a scheme $Y$ over $S$ together with $...
- 5,163
5
votes
2
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905
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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 ...
- 83
7
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3
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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
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3
answers
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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
4
votes
0
answers
167
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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
11
votes
1
answer
918
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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
13
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2
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1k
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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
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2
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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
4
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4
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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 ...
- 16k
25
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4
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4k
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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
16
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2
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2k
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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
19
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6
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4k
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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
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10
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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 ...
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3
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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