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Help with Macaulay2 computation of invariant ring

Consider the algebraic group $G:=\operatorname{SL}_{2}\times\operatorname{SL}_{2}$ acting on $V:=\operatorname{Mat}_{2\times 2}\oplus\operatorname{Mat}_{2\times 2}$ via the action $(A,B)\,\cdot\,(X,Y)=...
It'sMe's user avatar
  • 839
2 votes
0 answers
184 views

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 ...
Jon Elmer's user avatar
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2 votes
0 answers
172 views

Projective and Quasiprojective quotients

Let $G$ be a finite group acting on a projective variety $X$. Then $G$ also acts on $X-X^G$, where $X^G$ is the fixed locus. The GIT quotient varieties $X/G$ and $(X-X^G)/G$ are projective and quasi-...
Mark Shiffor's user avatar
2 votes
0 answers
107 views

Question about GIT: when is the map $\pi:X//_\theta G\rightarrow X//G$ birational?

I want to ask somebody who is more familiar with the theory of GIT quotients than I am, if there is a nice list of conditions on the action of a reductive group $G$ on an affine variety $X$ over ...
user42024's user avatar
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2 votes
0 answers
363 views

A simple problem on commutative algebra related to G.I.T

Let $G$ be a geometrically reductive algebraic group over an algebraically closed field $k$. Let $X$ be an affine variety over $k$ on which $G$ acts regularly. Then $G$ acts on the coordinate ring $A$ ...
Xin Nie's user avatar
  • 1,804
1 vote
0 answers
156 views

Software for computing invariant rings

I have an linearly reductive algebraic group $G$ acting regularly on an affine variety $X$(over an algebraically closed field of characteristic 0). I want to compute the invariant ring $\mathbb{K}[X]^{...
It'sMe's user avatar
  • 839
1 vote
0 answers
151 views

Generators of the same degree in a graded ring and GIT quotient

Let $S$ be a graded ring i.e. $S_d\cdot S_e \subset S_{d+e}$ where $S_d$ is the degree $d$ part of $S$. Assume $S_{<0}$ vanishes. For simplicity, you may think of $S$ as a graded subring of $\...
Hang's user avatar
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71 views

"Approximating" ring of semi-invariants

I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...
It'sMe's user avatar
  • 839