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Stable points in GIT: geometric picture

Is there a geometric picture justifying why "stable points" in GIT (Geometric Invariant Theory) are actually called "stable"? Stable, with respect to which effect? (Here, I have ...
JackYo's user avatar
  • 619
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)\...
Puzzled's user avatar
  • 8,998
5 votes
2 answers
905 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 ...
Turkelli's user avatar
5 votes
1 answer
605 views

Invariant section of a linearized sheaf

I am struggling to understand what an invariant section with respect to a linearization of a line sheaf is. In Geometric Invariant Theory, given a $k$-scheme $X$ (being $k$ an algebraically closed ...
Samantha Smith's user avatar
5 votes
1 answer
470 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 $G$-...
Anton Geraschenko's user avatar
5 votes
1 answer
343 views

About closed points in symmetric product schemes over a finite field

Let $k=\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation): Let $N$ be a positive ...
Lao-tzu's user avatar
  • 1,906
5 votes
2 answers
426 views

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 ...
Eleanor Von Hohlandsbourg's user avatar
5 votes
1 answer
450 views

Invariant ideal generated by invariant elements

Let $G$ be a complex reductive group acting linearly on $\mathbb{C}^n$ and let $X$ be a $G$-invariant closed subvariety of $\mathbb{C}^n$. Is $X$ the zero-set of finitely many $G$-invariant functions? ...
Simon Parker's user avatar
  • 1,383
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^...
CXbar's user avatar
  • 83
5 votes
1 answer
489 views

Fibre of GIT morphism

Let $ V $ be an affine variety (over $ \mathbb C$) with an action of a reductive group $ G$. I would like to consider the morphism $$ \pi : V \rightarrow V // G = Spec \, \mathbb C[V]^G $$ Let $ v \...
Joel Kamnitzer's user avatar
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 ...
SHP's user avatar
  • 779
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 ...
Jesko Hüttenhain's user avatar
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-...
It'sMe's user avatar
  • 839
5 votes
0 answers
246 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 ...
jg1896's user avatar
  • 3,328
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-...
User43029's user avatar
  • 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 ...
Xuqiang QIN's user avatar
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 ...
evgeny's user avatar
  • 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 ...
Yoyo's user avatar
  • 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 ...
stacksgg's user avatar
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 ...
guestmath's user avatar
  • 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 ...
guestmath's user avatar
  • 101
4 votes
4 answers
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 ...
Charles Siegel's user avatar
4 votes
2 answers
931 views

The closure $\overline{Gx}$ for an affine variety on which an reductive algebraic group acts

Let $G$ be a reductive group acting on an affine variety $X$. For simplicity, one may assume $G=SL_n$ or $G=U_n$ and assume the field is $\mathbb C$. Given this one can show $\mathbb C[X]^G$ is ...
Hang's user avatar
  • 2,789
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 ...
evgeny's user avatar
  • 1,980
4 votes
1 answer
245 views

Group action on affine variety induces faithful action on tangent space

I have a queestion about the proof of Lemma 2.2 from the paper arxiv 1105.3739: Let $G$ be a group acting faithfully on an irreducible affine variety $X=\operatorname{Spec}(A)$ over $k= \Bbb C$. ...
user267839's user avatar
  • 6,016
4 votes
1 answer
186 views

Are the two notions of free $\mathbb{G}_a$-actions equivalent?

Consider a finitely generated integral $\mathbb{C}$-domain $B$. An algebraic $\mathbb{G}_a$-action on $X:=\mathrm{Spec}(\mathcal{O}(X))$ is equivalent to a locally nilpotent $\mathbb{C}$-derivation $$\...
Yikun Qiao's user avatar
4 votes
1 answer
369 views

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 $\...
user avatar
4 votes
2 answers
757 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 ...
Jérémy Blanc's user avatar
4 votes
1 answer
279 views

Is quotient of projective scheme over arbitrary base by a finite group also projective

This question probably follows from standard geometric invariant theory. If true I'd to know a reference for it. Given a projective scheme $X\rightarrow S$ over the base $S$. Let's assume a finite ...
user127776's user avatar
  • 5,901
4 votes
1 answer
255 views

Symplectic structure of Higgs branch

I've been reading Kamnitzer's survey Symplectic resolutions, symplectic duality, and Coulomb branches. Here the Higgs branch is defined as a projective GIT quotient, but I couldn't figure out how the ...
Ji Woong Park's user avatar
4 votes
1 answer
255 views

Example of a line bundle not admitting a $\operatorname{PGL}(n+1)$-linearization in Mumford's GIT

$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Proj{Proj}\DeclareMathOperator\Pic{Pic}$I have a question about an example for a line bundle not admitting a $G$-linearization from Mumford's GIT, ...
user267839's user avatar
  • 6,016
4 votes
1 answer
257 views

Question regarding semistability of a point of GIT quotient

$\DeclareMathOperator\SL{SL}$I am currently looking at the paper titled "$\SL(2,\mathbb{C})$ quotients de $(\mathbb{P^1})^n$" by Marzia Polito. The author has considered diagonal action of $\...
tota's user avatar
  • 585
4 votes
1 answer
260 views

Question about valuation and blow up (a lemma in GIT book)

I'm reading Mumford's book Geometric Invariant Theory and confused about the proof of a lemma on Page 91&92: Lemma: Let $V_0$ be a smooth surface over an algebraically closed field $k$ with char$...
Kim's user avatar
  • 565
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{...
Dr. Evil's user avatar
  • 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 = \...
Libli's user avatar
  • 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 ...
Lewis Topley's user avatar
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 ...
Simon Parker's user avatar
  • 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 ...
user313212's user avatar
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 $U$ ...
a_g's user avatar
  • 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 $\...
Frosinoneculone's user avatar
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 ...
matthew's user avatar
  • 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 ...
Peter Crooks's user avatar
  • 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 ...
Bugs Bunny's user avatar
  • 12.4k
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 ...
Avi Steiner's user avatar
  • 3,079
3 votes
2 answers
367 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 ...
Jesko Hüttenhain's user avatar
3 votes
1 answer
320 views

Is the Hilbert Mumford Criterion true over the reals?

The Hilbert Mumford Criterion as in Wallach Theorem 3.24 says: Let $G$ be a linearly reductive subgroup of $GL(n, \mathbb{C})$. Let $(\sigma, V)$ be a regular representation of $G$. For a vector $v \...
Arielle Leitner's user avatar
3 votes
1 answer
1k views

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 $...
Wanderer's user avatar
  • 5,163
3 votes
1 answer
288 views

Question on geometric invariant theory

I'm reading Geometric Invariant Theory by Mumford, and confuse about the Proposition 2.4 on P54. It states that: Let $G$ be a reductive group, act on an algebraic scheme. Then the action of $G$ on $X$ ...
Kim's user avatar
  • 565
3 votes
2 answers
336 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 ...
IMeasy's user avatar
  • 3,779
3 votes
2 answers
543 views

Actions with finite stabilizer

Consider a affine variety $X$ over the field of the complex numbers, and an action of a reductive group $G$ on $X$ (I will consider the case of $G$ not finite, in particular $G=\mathbb{C}^*$). Reading ...
mathstudent's user avatar