All Questions
Tagged with ag.algebraic-geometry geometric-invariant-theory
195 questions
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 $...
- 15.5k
1
vote
0
answers
176
views
When the action of reductive group on algebraic variety is not equidimensional?
I saw the question When is an almost geometric quotient flat? which said
"The quotient $\pi$ is flat if and only if $\pi$ is equidimensional and $X$ is smooth".
I am curious is there an ...
- 63.9k
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 ...
- 1,906
1
vote
0
answers
144
views
Non-empty stable locus of an irreducible component
I have a vague question:
Let $X$ be an algebraic pre-scheme and $G$ be a linear reductive group. Consider the G.I.T. quotient $X{/\!/}G$. Is there any result (maybe in some special case) which tells ...
- 839
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 $\...
- 166
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 $G=SL(n+1)$ ...
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
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 ...
- 63.9k
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 ...
- 35.5k
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 ...
- 12.9k
0
votes
0
answers
129
views
Quotient $(V -S)/G$ is a quasi-projective variety for every closed $S \subset V$ with free $G$-action
I have a question about the statement of Remark 1.4 following on Thm 1.3 from Burt
Totaro's paper "The Chow Ring of a Classifying Space" (p. 4):
Let $G$ be a reductive group over a field $k$....
- 5,966
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 ...
- 35.5k
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....
- 35.5k
1
vote
0
answers
86
views
If algebraic group $G$ acts faithfully on a $G$-qp variety $X$, then $G$ has a Faithful projective representation
In Michel Brion's survey on Linearization of algebraic group actions
is stated in Examples 3.2.2.(iv) following claim p 17
without proof:
We fix an algebraic group $G$ over field $k$ (of arbitrary ...
- 5,966
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=\{...
- 8,529
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})$...
- 10.7k
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 ...
- 12.9k
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,...
- 443
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 ...
- 1,908
2
votes
1
answer
383
views
$G$-invariant morphism and coarse moduli spaces
Let $G$ be an algebraic group acting on $X$ (a finite type scheme on $k$).
A $G$-invariant $k$-morphism $f : X \rightarrow S$ is a map such that the following commute:
$\require{AMScd}$
\begin{CD}
G \...
- 921
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 ...
- 9,650
1
vote
0
answers
362
views
Invariant ring of linear algebraic groups
Let $G$ be a connected linear algebraic group. This question concerns Hilbert's 14th Problem for the adjoint action of $G$ on itself. Let $k[G]^G$ denote the algebra of regular functions on $G$ ...
- 2,751
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
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?
...
- 2,851
0
votes
0
answers
115
views
Equivalence between coactions and actions plus a linearization line bundle
Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all ...
- 123
2
votes
1
answer
151
views
Geometric quotients obtained by throwing away limits
Set-up: Consider the action of $\mathbb{C}^*$ on $\mathbb{C}^4$ defined as follows: $(t,(x,y,z,w))=(tx,ty,t^{-1}z,t^{-1}w)$. I know that the affine GIT quotient is equal to $$\phi: \mathbb{C}^4 \to \...
- 61
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$...
- 1,093
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 ...
- 8,529
2
votes
1
answer
249
views
Is there an $SL_n$-invariant functional on the space of rational functions on the projective space $\mathbb P^{n-1}$?
Let the group $SL_n$ act on the projective space $\mathbb P^{n-1}$ in the standard way (both defined over $\mathbb C$).
Is there an $SL_n$-invariant (linear) functional on the space of rational ...
- 3,772
3
votes
1
answer
249
views
Is the irreducible locus of the character variety a principal bundle in Zariski topology?
Let $\Sigma$ be a compact orientable surface and let $G$ be a reductive algebraic group (say, $G=\mathrm{SL}_n(\mathbb{C})$ for simplicity). The representation variety is
$$
X_G(\Sigma) = \mathrm{Hom}(...
- 8,529
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
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$ ...
- 841
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
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
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
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 \...
- 18.7k
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. ...
- 79.9k
1
vote
0
answers
153
views
Descent of projective bundles
A problem studied in GIT is the descending of vector bundles (or more in general coherent sheaves) to quotients.
It is a result of Kempf that whenever we have a vector bundle over a quasiprojective ...
- 297
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 ...
- 1,516
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 ...
- 63.9k
3
votes
1
answer
190
views
(Co)tangent sheaves to good quotients
Suppose given a variety $X$ over an algebraically closed field $K$, $\mathrm{char}K = 0$, equipped with an action of a reductive group $G$. Suppose also that $X$ admits a good quotient $p\colon X\to Y:...
- 2,305
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. ( ...
- 63.9k
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 \...
- 8,529
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 ...
CommunityBot
- 1
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 ...
- 8,529
0
votes
2
answers
597
views
Motivating the Quotient of an Algebraic Variety
Let $X$ be a variety with a $G$-action by an algebraic group.
My question refers to a motivating example from:
https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf
Here is the relevant ...
CommunityBot
- 1
3
votes
1
answer
254
views
Quotient of a Fano variety by a torus
We work over an algebraically closed field of characteristic zero. Let $X$ be a Fano variety, and $T\cong \mathbb{G}_m^r$ a torus acting faithfully on $X$.
I think we can canonically linearize the ...
- 6,995
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 ...
- 63.9k
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
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-...
- 63.9k