All Questions
26
questions
2
votes
0
answers
235
views
Action of algebraic group in cohomology of equivariant algebraic vector bundle
Let $X$ be a projective algebraic variety over an algebraically closed field. Let an algebraic group $G$ act algebraically on $X$. Let $\mathcal{F}$ be a $G$-equivariant vector bundle (or, more ...
- 21.3k
1
vote
0
answers
43
views
What is the kernel of the differential of the orbit-stabilizer map for nonsmooth stabilizers?
$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic ...
- 565
9
votes
2
answers
544
views
Quotients of schemes by connected groups
Let $X$ be a variety over $k$ where the characteristic of $k$ is zero. Let $G$ be a connected reductive group scheme acting freely and properly on $X$.
By the Keel-Mori theorem, the quotient $X/G$ is ...
- 635
3
votes
1
answer
296
views
Is an equivariant projective morphism equivariantly-projective?
Let everything be over $\mathbb{C}$. Consider two varieties $X,$ $Y,$ where $X$ is normal and $Y$ is affine,
having regular $\mathbb{C}^*$-actions and
a $\mathbb{C}^*$-equivariant projective morphism
$...
- 1,647
8
votes
0
answers
265
views
Generalization of a standard algebraic group theory result for a tensor problem
$\DeclareMathOperator\GL{GL}$Let $X$, $Y$, $Z$ be $\mathbb{C}$-vector spaces, and let $A\subseteq X$ and $B\subseteq Y$ and $C\subseteq Z$ be linear subspaces. Let $V=X \otimes Y \otimes Z$, acted on ...
- 970
0
votes
0
answers
114
views
Equivalence between coactions and actions plus a linearization line bundle
Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all ...
- 505
3
votes
1
answer
154
views
Weights on the linearization
Consider, just as an example, an action of $\mathbb{C}^*$ on $\mathbb{P}^2$ of the form
$$t\cdot p=[p_0:tp_1:t^2p_2]$$
There are $3$ fixed points, namely $e_1,e_2,e_3$. If I consider a $\mathbb{C}^*$-...
- 133
4
votes
0
answers
133
views
Linear vs algebraic unipotent quotient stacks
Consider algebraic stacks of the form $\mathbb{C}^n/G$ where $G$ is a unipotent group satisfying either
Type 1: $G$ acts on $\mathbb{C}^n$ via affine linear transformations
Type 2: $G$ acts on $\...
- 3,592
6
votes
1
answer
307
views
A class of unipotent group actions
Consider algebraic actions of unipotent groups $G$ on affine spaces $X=\mathbb{C}^n$. I am looking for a condition that would guarantee that the quotient $X/G$ exists and is also an affine space. For ...
- 3,592
0
votes
0
answers
367
views
Quotient of an affine scheme by an étale finite group
Let $G$ be a finite étale group scheme over a field $k$ and $X=\mathrm{Spec}(A)$ be an affine scheme on which $G$ acts. The categorical quotient $X/G$ exists and may be described as $\mathrm{Spec}(A^H)...
- 519
6
votes
3
answers
1k
views
Fixed points under a finite group action on projective variety
Let us have an algebraic action by a finite group G on a complex projective variety $X=\bigcup\limits_{i=1}^N X_i$, whose irreducible components $X_i$ are all smooth and of the same dimension $d$, and ...
- 1,647
5
votes
1
answer
109
views
(Euclidean) open orbit in an irreducible real algebraic set
Let $\tau:GL(n,\mathbb{R}) \rightarrow GL(V)$ be a rational representation of the general linear group of degree $n$ on a finite-dimensional real vector space $V$. Let $C$ be an irreducible real ...
2
votes
1
answer
151
views
G-sweep of irreducible sub variety
Let $G$ be a connected reductive algebraic group and $X$ be a $G$-variety. Let $Y$ be a $G$-invariant irreducible sub variety of $X$ which has non-trivial intersection with the semi stable locus $X^{...
- 113
8
votes
2
answers
919
views
Unipotent algebraic group action on quasi-affine (vs. affine) variety?
This question arises from a comment by user nfdc23 on an unrelated recent MO question here. It concerns textbook treatments of what has been called the "Theorem of Kostant-Rosenlicht", stated as ...
- 52.5k
7
votes
1
answer
438
views
Understanding a germ of a GIT quotient
Let $X$ be a smooth complex affine variety, let $G$ be a complex reductive group acting on $X$. Suppose that the stabilizer $G_x$ of a point $x\in X$ is reductive and connected. Let $\varphi: X\to X//...
- 14.1k