Questions tagged [geometric-invariant-theory]
for questions on geometric invariant theory (or GIT), including stability criteria and symplectic quotients.
242 questions
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Exists $G$-equivariant embedding with faithful representation of $G$?
Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT,...
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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 ...
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Relationship between vector bundles and modules
THE GROTHENDIECK RING IN GEOMETRY AND TOPOLOGY - M.F. ATIYAH
§1. The Grothendieck ring in homotopy theory
I am going to be talking about vector bundles, i.e. fibre bundles with
fibre a vector space ...
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Question regarding Hilbert scheme of points
$\DeclareMathOperator\SL{SL}$Let us consider $\SL(2,\mathbb{C})$ quotients of $(\mathbb{P^1})^n$ in the following sense. We consider diagonal action of $\SL(2,\mathbb{C})$ over $(\mathbb{P^1})^n$ ...
- 585
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Bondal-Orlov conjecture on Calabi-Yau varieties
Recently, I am trying to study the various progress made on the Bondal-Orlov conjecture: Birational Calabi-Yau varieties ⟹ Equivalent derived categories.
I have started reading the paper by Bridgeland ...
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164
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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 $\operatorname{GL}_n(\mathbb{C})$, ...
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On the trivialization of the sheaf of kahler differentials on the G-invariant topology
Let $X$ be a connected, smooth affine algebraic variety over an algebraically closed field $K$ of characteristic zero. Assume we have a finite group $G$ acting on $X$ by morphisms of $K$-schemes. ...
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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 ...
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Invariants of general linear groups under torus action
Let $G=GL_n$ be the general linear group (let's say over an algebraically closed field of char $=0$). Let's denote as $T$ the torus of diagonal matrices: is there an explicit description of the ...
- 1,061
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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 $...
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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 ...
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270
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In GIT, why are the semistable/unstable loci defined pointwise, instead of defining semistable/unstable subschemes?
Let $k$ be an algebraic closed charateristic zero field. Let $G$ be a reductive group over $k$ and let $X$ be a scheme (not necessarily separated) of finite type over $k$. Let $L$ be a line bundle ...
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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
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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 ...
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Nontrivial Invariants of trilinear functionals
The group $\operatorname{SL}(n_1,\mathbb{C}) \times \operatorname{SL}(n_2,\mathbb{C}) \times \operatorname{SL}(n_3,\mathbb{C})$ acts on ${\mathbb C}^{n_1} \otimes {\mathbb C}^{n_2} \otimes {\mathbb C}^...
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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 $\...
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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 $G=SL(n+1)$ ...
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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 $G$ ...
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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 ...
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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 ...
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Group actions on affine varieties with closed orbits
The following is motivated by a (now-deleted) MSE-question by @aglearner.
Suppose that $X\subset {\mathbb C}^n$ is an affine subvariety, equipped with the classical (Euclidean) topology. Consider the ...
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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 $k$....
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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 ...
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994
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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 $\mathrm{SL}_2(\mathbb{C})$...
- 2,751
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290
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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 ...
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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 ...
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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=\{...
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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-...
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$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 \...
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605
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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 ...
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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$ ...
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Why is the image of closed invariant subsets closed? Mumford, GIT, Theorem 1.1
I'm currently studying Mumford's Geometric Invariant Theory. Unfortunately, I'm stuck understanding a detail in Theorem 1.1.
(Partial) Claim of Theorem 1.1
Let $X = \operatorname{Spec} R$ be an affine ...
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147
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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$ (...
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450
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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?
...
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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 \...
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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$...
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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 ...
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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}(...
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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 ...
- 2,649
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Self duality and anti-self duality of Weyl curvature in four dimension
I am trying to compute explicitly in terms of extrinsic curvature the self dual part $W^+$ and the anti-self dual part $W^-$ of the Weyl tensor $W$ associated with a codimension 1 submanifold into ...
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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 ...
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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 $X$ ...
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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 ...
- 565
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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{\...
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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 ...
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GIT and singularities
Let $G$ be a complex reductive group acting on a complex affine variety $X$ and let $X // G = \operatorname{Spec}\mathbb{C}[X]^G$ be the GIT quotient.
Is there a relationship between the singular ...
- 1,383
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305
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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 \...
- 3,318
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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
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Jordan decomposition on the dual Lie algebra
$\newcommand\fg{\mathfrak g}\newcommand\gl{\mathfrak{gl}}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field, and let $G$ be a smooth, affine algebraic ...
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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. ...
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