16
votes
Accepted
Why is Mumford's GIT-quotient so effective?
By my understanding, your question is not "Why is Mumford's construction better than the affine quotient". As you note, Proj is better than Spec of invariants for taking quotients by $\mathbb{G}_m$. ...
- 148k
12
votes
Accepted
Standard Monomial basis for other types
Standard monomial theory has been extended to all classical groups by Lakshmibai, Seshadri and others in the series of papers "Geometry of $G/P$ I-IX".
A very concise description of standard tableaux ...
- 15.5k
12
votes
Accepted
Stable points in GIT: geometric picture
A point $x$ of a moduli space $\mathcal X$ is unstable if there is a map from $\mathbb A^1 \to \mathcal X$ that restricted to $\mathbb G_m$ is the composition of the constant map $\mathbb G_m \to \...
- 148k
11
votes
Accepted
Moduli of smooth curves in $|\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2,2)| $ and their invariants
[EDITED to exhibit $j$ as a rational function of $J_2,J_3,J_4$,
and to fix various local errors etc.]
The action of ${\rm SL_2} \times {\rm SL_2}$ on the $9$-dimensional space of
$(2,2)$ forms has a ...
- 79.9k
10
votes
Accepted
A duality result for Coxeter groups
Yes. By Chevalley-Shepard-Todd, $S(V)^G$ and $S(V)^H$ are polynomial rings. Let $S(V)^G=\mathbb{R}[g_1, \ldots, g_n]$ and $S(V)^H = \mathbb{R}[h_1,\ldots, h_n]$ where the $g_i$ and $h_i$ are ...
- 156k
9
votes
Accepted
Preparation for GIT (Geometric Invariant Theory)
Before I started reading/referencing Geometric Invariant Theory by Mumford, Fogarty, Kirwan I read Dolgachev's book Lectures on Invariant Theory. I also like Peter Newstead's book Introduction to ...
- 8,529
9
votes
Invariants of $\mathrm{GL}_n$ representations
When $k=1$ the ring of invariants of degree $d$ separately in each of $W = \mathrm{Sym}^2 V$ and $W^\star = \mathrm{Sym^2} V^\star$ has dimension equal to the number of partitions of $d$ with at most $...
- 11.2k
8
votes
Accepted
GIT quotient vs. largest Hausdorff quotient
I recommend reading the very nice exposition here (which is where I learned this fact):
Schwarz, Gerald W. The topology of algebraic quotients. Topological methods in algebraic transformation groups (...
- 8,529
8
votes
Accepted
Vector bundles on quotient variety
I put my comment as an answer: the necessary and sufficient condition for $E$ to be the pull back of a vector bundle on $X/G$ is that the stabilizer of any closed point $x$ with a closed orbit
acts ...
8
votes
Accepted
Variety of commuting matrices
There's been a good deal of work since the papers Mark Grant cited.
The rational cohomology of $\mathrm{Hom}(\mathbb{Z}^n,K)//K$ for $K$ a compact connected Lie group was computed by Stafa (https://...
- 8,803
8
votes
Character variety of the free group
I think it depends on what you want.
The $\mathrm{SL}_2(\mathbb{C})$-character variety of a free group $F_r$ admits a model over $\mathbb{Z}$ (maybe its better to say $\mathbb{Z}[1/2]$); see Rank 1 ...
- 8,529
7
votes
Accepted
An explicit negative solution to the Lüroth problem for non-algebraically closed fields
According to the first paragraph in Shafarevich's paper "On Luroth's problem" (found here http://www.math.ens.fr/~benoist/refs/Shafarevich.pdf) the field of rational functions on the surface ...
- 18.7k
7
votes
Accepted
Is there a Chevalley map for spherical varieties?
Edit: The answer to question 1 is yes if $G/H$ is a symmetric variety as the OP pointed out.
For arbitrary spherical varieties the answer is no in general. If my memory serves me right, the spherical ...
- 15.5k
7
votes
Accepted
Are the two notions of free $\mathbb{G}_a$-actions equivalent?
No, they are not. A nice example has been constructed by Winkelmann in "On free holomorphic $\mathbb C$-actions on $\mathbb C^n$ and homogeneous Stein manifolds", Math. Ann. 286 (1990), 593–...
- 15.5k
7
votes
Stable points in GIT: geometric picture
I am posting an answer that expands my comment. The posted question could mean, "What is the essential aspect of 'stable points' to a mathematician today?" However, it could also mean, &...
7
votes
Accepted
Group action on affine variety induces faithful action on tangent space
Suppose that $k$ is a positive integer, and we have shown that $m_x^k$ equals $V^k + m_x^{k + 1}$. Then $m_x^{k + 1}$ equals $(V + m_x^2)(V^k + m_x^{k + 1}) = V^{k + 1} + (m_x^2 V^k + (V + m_x^2)m_x^{...
- 12.9k
6
votes
Accepted
The closure $\overline{Gx}$ for an affine variety on which an reductive algebraic group acts
The answer to (2) is affirmative: an affine $G$-variety with a dense open orbit (like $\overline{Gx}$) contains a unique closed orbit. The reason for that is that any two closed are orbits are ...
- 15.5k
6
votes
Is the conjugation action linearizable?
Sumihiro proved that every normal quasi-projective $k$-scheme with an action of a smooth affine connected group $G$ has an ample $G$-equivariant line bundle. This applies in particular to any action ...
- 8,972
6
votes
Accepted
GIT and singularities
This is an answer to the revised question. It is the simplest counterexample that I can think of where the reductive group is smooth and connected, where $X$ is normal and affine, and where $Y=X//G$ ...
6
votes
Is the Hilbert Mumford Criterion true over the reals?
The theorem holds over any perfect field $k$ and for any affine $G$-variety $X$ defined over $k$. More precisely, it states: Let $x\in X(k)$ be a $k$-rational point and $Y\subseteq\overline{Gx}$ be a ...
- 15.5k
6
votes
Stable points in GIT: geometric picture
The Kempf-Ness theorem point of view on GIT roughly says that there is a function (the norm of a vector in a representation in their paper but there are other functions that work in different examples)...
- 3,409
5
votes
Cohen-Macaulay rings in GIT
Cohen-Macaulayness is instrumental in GIT when it comes to counting invariants of a given degree.
Let $R=\bigoplus_{d\ge0}R_d$ be a finitely generated graded $\mathbb C$-algebra (for example a ring ...
- 15.5k
5
votes
Accepted
Question on geometric invariant theory
I believe this is addressed on page 52, underneath the statement of Iwahori's theorem. He provides an argument for why Iwahori's result can be strengthened to G reductive by considering $G \rightarrow ...
- 841
5
votes
Variety of commuting matrices
My new paper with Carlos Florentino and Jaime Silva answers this question:
Mixed Hodge structures on character varieties of nilpotent groups.
In particular, see Section 4.4.
For an implementation of ...
- 8,529
5
votes
Accepted
Character variety of the free group
Let $\pi$ be a free group of rank 2. The character variety of $SL_{2,k}$-representations of $\pi$ is always isomorphic to affine 3 space $\mathbb{A}^3_k$ for any ring $k$.
Let $A[\pi] = A[\pi]_k$ ...
- 10.7k
5
votes
Accepted
About closed points in symmetric product schemes over a finite field
There are probably many ways to answer your question depending on what your preferred point of view on schemes and symmetric products are. Let me offer the following approach.
Forget that $k$ is a ...
- 32.4k
5
votes
Accepted
Invariant ring of the subvariety
As we discussed in the comments, by linear reductivity of $G$, there is a $G$-module splitting of the surjection $\mathbb K[X] \to \mathbb K[Y]$; so the restriction map $\mathbb K[X]^G \to \mathbb K[Y]...
- 12.9k
5
votes
An easy textbook for geometric invariant theory and moduli space which makes use of scheme theory
The OP asked me about the mistake in Mumford's book (probably well-known to experts). I am attaching below something I wrote about this more than 10 years ago.
Dear Johan and Jarod,
At Stony Brook ...
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