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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Understanding the local structure at a point on the moduli of semistable bundles of rank 2 of fixed determinant over a curve

First, let me describe the premise. Let $X$ be a smooth projective curve of genus greater than 2 over $\mathbb{C}$, let $x,y\in X$ two closed points, and let $M_{y-x}$ denote the moduli space of S-...
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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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Encoding the position information in real distance metric

The Euclidean distance doesn't preserve the exact position information. For example, the distance of the points (3,1) and (1,3) would be the same from the origin. Is there any distance metric which ...
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Reference on Geometric Invariant Theory

I am currently self-learning Geometric Invariant Theory on Mumford book, but found it is really hard to me. Is there any good reference for self-study? (Algebraic geometry flavour is better, although ...
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Some basic conceptual questions regarding Algebraic Group.

There is mention of the statements like the following which appears in every lecture notes or expository articles regarding algebraic group (or may be GIT). The first statement is more or less as ...
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Intuition for semistable points on a $G$-variety.

I'm reading some lecture notes on the relationship between GIT quotients and symplectic reduction, and came across the definition of a semistable point. For completeness, I will re-write it below: ...
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Image of a subvariety under algebraic group actions

Consider a linear algebraic group $G$ acting on an affine variety. I am interested in knowing some information about the following two questions: 1) Is there a subvariety whose image under $G$ is the ...
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118 views

Largest subgroup in which a given polynomial is invariant.

I am trying to solve the following question; Given a polynomial $f\in \mathbb{C}[x_{1},x_{2},\ldots,x_{n}]$, find the largest subgroup $\Gamma\le GL(\mathbb{C}^{n})$ such that $f\in \mathbb{C}[x_{1},...
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117 views

Moment maps unitary group acting on matrices

I am reading the fifth chapter of "An introduction to extremal Kaehler metrics" by Gabor Szekelyhidi. At the very beginning of that chapter, the author describes moment maps and Hamiltonian action. ...
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Does a one-parameter subgroup of a reductive group 'act reductively'?

For the present question I am only interested in the case where the underlying field is $\mathbb{C}$. Hilbert fourteenth can be stated as follows: Let $V$ be an algebraic variety and $G$ a linear ...
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114 views

GIT-Quotient of $\mathbb{C}^*$ acting on $\mathbb{C}^{m+1}$

I'm new to GIT and I came across the easy example of $G=\mathbb{C}^*$ acting on $V=\mathbb{C}^{m+1}$ by scalar multiplication, given a character $\chi(t) = t^k$ of $\mathbb{C}^*$ for some integer $k$. ...
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214 views

Does this categorical quotient exist?

Look at the example 3 in the following picture and also its orbits. It is from the book: Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory by A.N. Parshin,I.R. Shafarevich. The ...
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Canonical forms of the natural action of $O(n,\mathbb{C})$ on $sym(n,\mathbb{C})$

I want to learn about the natural action of the complex orthogonal group, $O(n,\mathbb{C}):=\{g\in M(n,\mathbb{C}) : gg^T=\mbox{Identity}\}$, on the complex vector space of complex symmetric $n\times ...
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How to compute $ \mathbb{C}[p_1, q_1, p_2, q_2]^{\mathcal{S}_2}$ under $(p_1, q_1) \leftrightarrow (p_2, q_2)$?

Let $B_1 := \mathbb{C}[p_1, q_1], B_2 := \mathbb{C}[p_2, q_2]$ , and $(B_1 \otimes B_2)^{\mathcal{S}_2} = \mathbb{C}[p_1, q_1, p_2, q_2]^{\mathcal{S}_2}$. Here, the action of $\mathcal{S}_2$ is $(p_1, ...
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Stability and the Hilbert Polynomial of an Image

In Huybrechts and Lehn it says that for two semistable sheaves $\mathcal{F},\mathcal{G}$ and a sheaf homomorphism $f : \mathcal{F} \to \mathcal{G}$, writing $$\mathcal{E} := \text{im}(f)$$ yields the ...
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191 views

Properties preserved by categorical quotient (after Mumford's GIT)

A categorical quotient of a variety $X$ acted by $G$ is a morphism $f: X \to Y$ constant on $G$-orbits such that every $h: X \to Z$ constant on $G$-orbits factors through it. Mumford on the page 5 of ...
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Physicist trying to understand GIT quotient

I am reading Nakajima's textbook on Hilbert Schemes. I am trying to understand some very basic facts about the GIT quotient. We start with a vector space $V$ over $\mathbb{C}$. Let $G \subset U(V)$ ...
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Quotient and Cohomology

Let $G=\mathbb Z_2$ and let $X \subset \mathbb P^5$ be a projective variety and the action of $G$ on $X$ is given by $1.(x_1,x_2,x_3,x_4,x_5,x_6)=-(x_6,x_5,x_3,x_4,x_2,x_1)$. I need to compute the ...
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Proj of an algebra

Let $\mathbb Z_2= \langle\sigma\rangle$ act on $\mathbb C^6$ by $(x_1,x_2,x_3,x_4,x_5,x_6)=-(x_6,x_5,x_3,x_4,x_2,x_1)$. Then what is $\operatorname{Proj}\left(\left(\frac{\mathbb C[x_1,x_2,x_3,x_4,x_5,...
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Linear non-equivalence of regular ternary cubics in normal form

Consider $G:=\operatorname{GL}_3(\Bbb C)$ acting from the right on $R=\Bbb C[x_0,x_1,x_2]_3$ by linear substitution, i.e. we let $x=(x_0,x_1,x_2)^T$ and the action of $A\in G$ on $f\in R$ yields the ...
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170 views

Is affine GIT quotient necessarily an open map?

Let $k$ be a field, $X=$Spec$A$ be an affine scheme with A a f.g. $k$-algebra. $G=$Spec$R$ is a linearly reductive group acting rationally on A. (i.e. every element of $A$ is contained in a finite ...
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279 views

Intuition behind definition of Stable Bundles?

To my understanding, given a rank $r$ and degree $d$, we can fix a $C^{\infty}$ vector bundle $\mathcal{V}$ over some curve $\Sigma$ of genus $g$. We then want to study the moduli space $M_{g}(r,d)$ ...
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1answer
220 views

Non-separated quotient of separated scheme

I am reading Mumford's GIT book. I found the following claim there. Let $X$ be an algebraic variety. Let $G$ be an algebraic group acting on $X$. Then the categorical quotient of $X$ by $G$ may be ...
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Regular functions on $Proj(S)$

Let $S=\bigoplus_{d\geq 0}S_d$ denote a graded algebra over a field $k$ (say S integral, finitely generated, and $k$ algebraically closed). Under which conditions does $\mathcal{O}(Proj(S))\simeq S_0$ ...
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370 views

Two equivalent definitions of GIT semistable points

Let $X$ be a projective variety, acted on by a reductive algebraic group $G$. We fix a linearization given by the $G$-equivariant ample line bundle $L\to X$. I am aware of two definitions of the ...
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151 views

Definition of the $\Bbb C^*$-weight of a line bundle

I'm new to geometric invariant theory and am unsure about a definition. Let $X$ be a smooth projective-over-affine variety equipped with a $\Bbb C^*$ action and let $Z$ be the fixed locus of this ...
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163 views

Homotopy groups relating to toric varieties

It is known that the toric variety $X_\Sigma$ of a simplicial fan $\Sigma$ can be constructed as a quotient $$X_\Sigma = \bigl(\mathbb C^N \setminus V(B)\bigr)/G.$$ Here $N$ is the number of rays, $B$...
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164 views

Linearization of a group action: why the map is equivariant?

I'm using Dolgachev's book on invariant theory to learn linearizations of group actions. Here is a sketch of main construction: let linear algebraic group $G$ act on a quasi-projective variety $X$, ...
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Geometric intuition for linearizing algebraic group action

I am reading an overview of geometric invariant theory and find myself stuck when we begin linearizing the action of an algebraic group on a variety. The definition given in my notes is that given an ...
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402 views

The unique closed orbit in GIT quotient fibers for polynomial actions of Gl

The following reasoning must contain a flaw somewhere because I end up with something absurd, and I cannot figure out where the mistake is. I hope that someone can point it out to me. Let $M$ be the ...
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1answer
248 views

Why the GIT quotient $\mathbb{A}^m//\mathbb G_{m}$ is empty?

Let $\mathbb{A}^m ={\text{Spec }}\mathbb{C}[x_1,\dots,x_m]$, and the multiplicative group $\mathbb G_m \cong \mathbb{C}^*$ acts on $\mathbb{C}[x_1,\dots,x_m]$ by $$\lambda_t (x_1,\dots,x_m)= (t^{a_1}...
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232 views

GIT quotient for a certain torus action on an affine space

I'm reading various books and some notes and here is my question. Let $(\mathbb{C}^*)^2$ act on $\mathbb{C}^4$ by $$(\lambda_1,\lambda_2).(x_1, x_2, y_1,y_2)=(\lambda_1 x_1, \lambda_2 x_2, \...
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204 views

A condition in the definition of geometric quotient

I am reading the first several pages of GIT by Mumford, and I need some clarification of one requirement in the definition of geometric quotient (c.f. Definition 0.4, GIT): Suppose a group scheme $G/...