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.

36 questions
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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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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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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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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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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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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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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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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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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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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 ...