# Tagged Questions

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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### How does grade projection act on homogeneous multivectors in geometric algebra?

I'm reading Clifford Algebra to Geometric Calculus by Hestenes, and struggling with an early result about reversion inside of a grade-projection operator. It is noted that $A_r$ and $B_s$ are ...
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### Proj of some graded algebra

I was computing some GIT quotients and came up with the following question: to compute $\mathrm{Proj}(\mathbb C [f_1,f_2,f_3,f_4,f_5,f_6]/I)$ where $f_i$'s are homogeneous polynomials of same degree ...
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### A criterion for orbits of complex reductive group to be closed

I am having some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following: Let $G=K_{\Bbb C}$ be a ...
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### Kähler quotients of affine varieties and GIT

Let $X\subseteq \Bbb C^n$ be a smooth affine variety and $G=K_{\Bbb C}$ a complex reductive group acting linearly on $\Bbb C^n$ preserving $X$ (where $K$ is a maximal compact subgroup of $G$). Suppose ...
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### From algebraic group actions to group scheme actions

I am trying to understand the basic results of geometric invariant theory. I want to pull off the band aid and use Mumford, but am a neophyte with respect to scheme theory. Thus, I have been trying to ...
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### Finding relations between invariant polynomials

Suppose I have an action of a linear reductive group ($GL(2,\mathbb{C})^2$ in this case) on a complex vector space (of dimension $16$) and I want to compute explicitly the ring of invariants of this ...
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### when does one want to use the Reynolds operator in GIT?

The role of Reynolds operator in GIT has always been a little mystery to me. Actually I see that in some proofs it gets used in an efficient way, but what I cannot grasp is the general philosophy. I ...
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### Scaling-Invariant Orbits of Semisimple Group Representations

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Note that if $V$ is the adjoint representation of $G$, then ...
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### When does a group action on a k-algebra induce an algebraic action on the spectrum?

This question arose from my last question, which I considered answered - from the comments, however, it is obvious that the answer is only complete in characteristic zero, and I am trying to ...
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### Partial (or complete) flag varieties as GIT quotients of affine spaces

I am looking for presentations of partial or complete flag varieties as GIT quotients of affine varieties spaces. That is, for a choice of of dimensions $0=d_1<d_2<\dots<d_k = n$, I would ...
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### Coarse moduli spaces of quotient stacks

Suppose you have a separated Deligne Mumford quotient stack $[V/G]$ over a field of characteristic $0$, where $V$ is a quasiprojective variety and $G$ is an algebraic group that does not necessarily ...
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### Normalization of quotient stacks

Suppose you have a Deligne Mumford stack which is a quotient $[X/G]$ of a scheme $X$ by an algebraic group $G$ . What is the normalization of that? Is it true that its normalization is a quotient ...
Let $C$ be a connected smooth curve, $0\in C$ a closed point and $W\rightarrow C$ a family of projective schemes. Assume that the fibers $W_t$ of $W$ are smooth for all $t\neq 0$ and that $W_0=Y_1\cup ... 2answers 461 views ### When does a G-invariant one to one map between two closed algebraic G-set descend to a one to one map on the G.I.T quotient ? I do not know much about Geometric Invariant Theory. My question is the following: Let$X$and$Y$be two complex affine or projective varieties. Let$G$be a reductive group which acts on both$X$... 2answers 209 views ### blow up of segre primal and$\mathcal{M}_{0,6}$The segre cubic primal$X\subset P^4$is the GIT quotient of 6 points on$P^1$. Let$M_{0,6}$the DM compactification of the moduli of 6-pointed rational curves. The Segre primal$X$is a cubic 3-fold ... 2answers 320 views ### degrees of the invariants for the action of$SL(V)$on$\wedge^4V$How can we find the degrees of the invariants for the action of$SL(V)$on$\wedge^4V$,$dimV=8$by the model in the Lie algebra$E_7$. 2answers 405 views ### Quotient of affine space by cyclic permutation The quotient of the affine space$\mathbb{A}^n$by the symmetric group$Sym_n$is again an affine space of the same dimension, and invariants are given by elementary symmetric polynomials. What ... 1answer 277 views ### Algebraic closure and GIT Does one need to work over an algebraic closed field in ordre to construct GIT quotients à la Mumford? If yes, why? 1answer 243 views ### What does this particular geometric quotient locally look like? Let$k$be a field and consider the algebraic group$GL_n$over$Spec(k)$. It has as a closed (but not normal) algebraic subgroup the group$M$of monomial matrices, i.e. matrices having exactly one ... 2answers 244 views ### Intersection theory for$G$-varieties - an action on the chow ring? Let$G$be a reductive algebraic group. Let$X$be a$G$-variety and consider any closed subvariety$Z$of$X$. Since any$g\in G$acts as an automorphism, we know that$g.Z$is again a closed ... 1answer 708 views ### A line bundle that does not admit a G-linearisation I have been thinking about quotients lately and pondered the following: Let$G$be a connected linear algebraic group and$X$a$G$-variety acting via the morphism$\sigma:G\times X\rightarrow X$. ... 3answers 587 views ### Algorithms in Invariant Theory Let$V$be a polynomial representation of the general linear group$\Gamma:=\DeclareMathOperator{\Gl}{Gl}\Gl_n(\newcommand{\C}{\mathbb C}\C)$. In chapter 4.6 of his book "Algorithms in Invariant ... 2answers 342 views ### When is an orbit spherical? I asked the following question over at math.stackexchange, but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here: Let's assume we have an affine, reductive, ... 1answer 242 views ### Lift of a morphism between geometric quotients Let$S$be a scheme. Definition. Let$X$be an$S$-scheme and$G$a smooth affine group$S$-scheme acting on$X.$An$S$-scheme$Y$is a geometric quotient of$X$by$G$if there exists a morphism$\...
Let $G$ be a reductive, affine, algebraic group over $\newcommand{\C}{\mathbb C}\C$. Let $X$ be a $G$-variety. For $x\in X$, we write $$G_x:=\{ g\in G\mid g.x=x\}$$ for its stabilizer and for any ...