Questions tagged [geometric-invariant-theory]

Geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces.

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Stability of vector bundles as GIT quotient

I believe that the stability of vector bundles or coherent sheaves (defined as the inequality of 'slopes' of its subsheaves) comes naturally from GIT. However, in any literature I can find, it is ...
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Certain GIT quotients for elliptic curves

The base field $k$ is algebraically closed. Let $C$ be a smooth projective variety and $G$ be a finite group of automorphisms of $C$ (as an algebraic variety). As is well known, up to isomorphism ...
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Automorphisms of curves and Hurwitz-Riemann formula

Let $k$ be a base field algebraically closed and of zero characteristic. Let $C$ be a smooth projective curve and $G$ a finite group of automorphisms of it. Let $C*$ be a smooth projective curve whose ...
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Dimension formula/algorithm for quiver varieties?

The title says the question. For a quiver and a dimension vector and a stability vector, we can construct a moduli space of semistable quiver representations with the given dimension vector. The ...
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Semistability and Proj quotients

Before to make my question, I'm going to introduce all necessary notions. Let $G$ be a linearly reductive group (i.e. each rational representation is completely reducible) acting regularly on an ...
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Geometric quotient parametrizing points which are not stable

I'm reading some lecture notes about GIT, and I was wondering the following: Let $\mathbb{C}^*$ act on $\mathbb{C}^2$ as $t\cdot (x,y)=(tx,t^{-1}y)$. One may show that there are 3 types of orbits: ...
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Semistable points on a hermitian vector space

I am reading Lectures on Hilbert schemes by Nakajima, and this one argument really confuses me. For context I will lay out what is currently happening: Let $V$ be a hermitian vector space, and $G$ a ...
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GIT stability of plane curve

How do I show that a plane curve of degree d is unstable ( in the GIT sense ) if it has a singular point of multiplicity > 2d/3? This is an exercise from Dolgachev’s book Lectures on invariant ...
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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 $U$...
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How to determine if an invariant rational function is defined at the $\theta$-polystable point?

Background: Let $A$ be a finite-dimensional (associative and unital) algebra over $\mathbb{C}$. Assume there is a quiver $Q=(Q_0,Q_1)$, where $Q_0$ is the set of vertices and $Q_1$ is the set of ...
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Sufficient condition for moduli space of slope-stable bundles to be non-empty

I'm trying to find to a statement concerning the non-emptiness of the moduli space of slope stable vector bundles over a Kähler surface in the literature. Let $X$ be a Kähler surface. Let $\mathscr{M}(...
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invariant closed subset under group scheme action (Mumford GIT)

I have a question about a condition in Mumford's GIT book. One page 8, remark (6) iii) they talk about a closed subset $W$ of $X$ which is invariant under the action of $G$. The context is that $G$ is ...
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Why is this point fixed?

For an algebraic group $G$ that acts by $\sigma$ on a proper algebraic scheme $X/k$ and some closed point $x\in X$, we define the morphism $\psi_x: G\to X$ by $g\mapsto g\cdot x$. Say $\lambda$ is a $...
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Why $\mathbb{A}_K^n/G_n=\mathbb{A}_K^n$?

In my algebraic geometry class my professor mentioned that $\mathbb{A}_K^n/G_n=\mathbb{A}_K^n$, where $G_n$ is the group of permutation of degree $n$ acting on coordinates. This is not clear for me ...
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Relationship between GIT and coarse moduli spaces

I'm trying to understand how a generic algebraic geometer constructs coarse moduli spaces. I'm familiar with the definition, and how it is usually quite involved to show that a space has the coarse ...
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Computing the stabilizer of any polynomial of degree $n$ with distinct roots.

How to compute the stabilizer of any polynomial of degree $n$ with distinct roots? I was told in case of polynomials of degree 3 in 2 variables this is related to the fact that there is a unique ...
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Proof for "any $p \in \operatorname{Sym^3(\mathbb C^2)}$ can be written as a product of linear factors?"

Here is the paper I am trying to understand: I need a proof that "any $p \in \operatorname{Sym^3(\mathbb C^2)}$ can be written as a product of linear factors $(a_1 x + b_1 y)(a_2 x + b_2 y)(a_3 ...
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Why are the linear factors of $g.p$ are given by $(a_i, b_i)g^{-1}$?

I am reading a part of the paper below that computes the semistable locus in case of $\operatorname{Sym^3}(\mathbb{C}^2).$ Here is the part of the paper I do not understand: Specifically, I do not ...
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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 $\implies$ Equivalent derived categories. I have started reading the paper by ...
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Does the dualizing sheaf on the nodal rational curve not admit a $\mathbb G_m$-linearization?

Let $X$ be the projective variety obtained by gluing the two points $0, \infty \in \mathbb P^1$ transversally, which means that $X$ is isomorphic to $\{y^2z = x^2(x+z)\} \subset \mathbb P^2$, the ...
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inverse image of the symmetrization map

Let $S_n$ denote the vector space of $n \times n$ matrices and $M_{m \times n}$ denote the vector space of $m \times n$ matrices. Then we have a map $f: M_{m \times n} \times M_{n \times m} \...
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Non-empty stable locus in 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 us ...
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Actions by affine algebraic groups?

I am learning geometric invariant theory from Hoskins' notes. I am really confused about notions of group actions. One particular problem is the definition of a (rational) action on an algebra. In ...
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Geometric quotient definition in Mumford's book contains an error?

The seminal book by Mumford titled "Geometric Invariant Theory", 1965, gave a thorough and satisfactory answer to the question of how to form quotients of schemes. Not only that, it laid the ...
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Linearization of the action of PGL(n) on the projective space of dimension n-1

Since I do not have enough reputations I could not comment directly at a similar question asked at https://mathoverflow.net/questions/219064/choosing-a-group-action-to-do-git-of-hypersurfaces?newreg=...
chi-yu Cheng's user avatar
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Global sections of $\operatorname{Proj} S$ when $S$ is an integrally closed domain

Let $S=\bigoplus_{n \geqslant 0}S_n$ be a finitely generated $\mathbb{Z}_{\geqslant 0}$-graded algebra over $\mathbb{C}$ with $S_+ \neq 0$, here $S_+:=\bigoplus_{n>0}S_n$. Let us also assume that $...
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About Jordan normal form and stability in GIT

I am preparing linear algebra exam and I met a problem about Jordan normal form. Suppose $V$ is a vector space over field $\mathbb K$, $\psi$ is a linear transform on $V$, the char polynomial and ...
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What if the semi-invariant ring is a polynomial ring or hypersurface

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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What is Representation of Surface Groups?

I had a question in my mind for a month ago. Mainly I am interested in Hyperbolic Geometry. I found a topic named "Representation Theory of Surface Groups". Let me tell about what is a "...
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Proving that GIT quotient of linear group is smooth

I'm trying to prove that the projective variety associated to the GIT quotient $$\mathrm{End}(W)^{ss} / H$$ is smooth, where $H$ is a (reductive) subgroup of $\mathrm{Gl}(W)$ acting by pre-composition ...
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Understanding (GIT) quotient by action on End(W)

Suppose we have a subgroup $H < \mathrm{Gl}(W)$, where $W = \mathbb{C}^n$, which acts on $\mathrm{End}(W)$ by precomposing: $a \mapsto h\circ a$, for $h\in H$. We are given a distinguished linear ...
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Why are finite groups linearly reductive?

Let $G$ be a linear algebraic group contained in $GL(n)$. $G$ is linearly reductive iff every regular representation is completely reducible. Among the examples of linearly reductive groups, there are ...
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Example of G-invariant ideal

Definition: Let $G$ be a group acting on $R_n:=K[x_1,\dots,x_n]$ with $$\begin{aligned} G\times R_n &\rightarrow R_n\\ (g,f) &\mapsto f^g \end{aligned} $$ where $(f^g)$ acts on a point $p$ in ...
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Help with an algebraic geometry result

I am studying this part of algebraic geometry and I have come to this proposition. I understand the basic idea well but there are two details that escape me. Proposition: Let $G$ be a finite group ...
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Reference for the Hilbert scheme of points as GIT quotient

Could you provide me some ressources concerning the construction of the Hilbert scheme of n ponts in $\mathbb{C} ^2$ as a GIT quotient. There is this article of Barbara Bolognese and Ivan Losev but ...
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Action of an algebraic group is closed.

I am studying geometric invariant theory and I came to the following question. Let us consider an algebraic subgroup $G$ of $SL(V)$ with $V$ a $n$-dimensional $k$-vector space. Let us consider the ...
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Relative GIT quotient.

I´m studying Mumford's geometric invariant theory and I came to the following question. Let us suppose that $k$ is an algebraically closed field of characteristic 0, and $G$ is a reductive linear ...
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Quotient of closed $G$-invariant subset of $G$-variety

Let $X$ be an affine $G$-variety where $G$ is a reductive group. All the varieties are over $k$ , where $k$ is a field (if it necessary we can assume it is algebraically closed). It is a known theorem ...
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Invariants cyclic group actions

Let $G_m$ be the multiplicative group of $m$-th roots of unity generated by $\epsilon_m=\exp{(2\pi i/m)}$, and let us assume it acts faithfully on $\mathbb{C}[x_1,x_2]$ with weights $a=(a_1,a_2)=(a_1,...
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Is the product group of finitely many copies of the multiplicative group of positive real numbers a reductive group?

Let G be the multiplicative group of positive real numbers. Is the finite product group $G \times \cdots \times G$ reductive? I am trying to construct the moduli space for some quiver representations ...
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Quotients of $\mathbb{P}^1 \times \mathbb{P}^1$

It is known that $\mathbb{P}^1 \times \mathbb{P}^1 \not \cong \mathbb{P}^2$. One way to see this is working with their respective class groups, $\mathbb{Z}^2$ and $\mathbb{Z}$, which in this case, ...
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All Harder-Narasimhan factors of $E$ are semistable with slope $\leq B \in \mathbb{R}$ implies $E$ semistable

I'm trying to show the following claim: Let $E$ be a vector bundle on a surface $X$, of slope $\mu(E) = B$ such that it's Harder-Narasimhan factors are $\mu$-semistable of slope $\leq B \in \mathbb{...
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Proof of "A geometric quotient is categorical"

I'm reading Geometric Invariant Theory by Mumford-Fogarty, but I can't understand some details in the proof that any geometric quotient is categorical. Let $\sigma$ be an action of $G/S$ on $X/S$ ...
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$G$-invariant divisor in affine space

Suppose we have a linearly reductive group $G$ acting on $X=\mathbb{A}^n_{\mathbb{C}}$ and we have a closed subvariety $Z=V(f_1,...,f_k)$, where $f_i$ are $G$-invariant functions. If we further know ...
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2-Beltrami derivative of the normal vector of surface in $\textbf{R}^3$

Assume a surface $\textbf{S}\in C^3$, $\overline{x}=\overline{x}(u,v)$, $(u,v)\in D$ and a function $f(u,v)\in C^2$ defined on $D$. Moreover let $\{\overline{e}_1,\overline{e}_2,\overline{e}_3\}$ be ...
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Group schemes vs abstract groups in GIT

In studying GIT I encounter the same problem at multiple occasions of confusing group schemes for abstract groups . There is a natural example: Take an affine scheme $X=Spec(A)$ and an (affine) group ...
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Stable points in a GIT quotient

I have a maybe stupid question on GIT: Let $\mathbb P^N=\mathbb P^N_k$ be the Hilbert scheme of hypersurfaces of degree $d$ in $\mathbb P^n$, where $N=\binom{n+d}{d}-1$, and let $G:=PGL_{n+1}(k)$. ...
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GIT quotients under equivariant blowup

I am looking for a reference. Let $X$ be a (complex projective) variety and $V$ some subvariety invariant under the action of a reductive group $G$. Has it been studied in general how blowing up $X$ ...
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What is "locally integral" in Mumford's GIT book?

I think "locally integral" means for each point of a scheme, its local ring is a domain. In my definition, "normal" obviously implies "locally integral". But "normal" and "locally integral" is ...
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Use of GIT for moduli problems

Was there an actual use of GIT technics (moment map, Kempf-Ness theorem) in the different proofs of the Kobayashi-Hitchin correspondence (by Uhlenbeck-Yau and Donaldson)? Or this was rather a ...
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