All Questions
Tagged with ag.algebraic-geometry algebraic-groups
914
questions
3
votes
1
answer
144
views
Extending Tannakian "dictionary" to gerbes
The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories".
Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the ...
- 531
6
votes
1
answer
234
views
Extensions of algebraic groups and extensions of fpqc sheaves
There are at least two ways to define $\operatorname{Ext}^p(F,G)$ when $F$ and $G$ are commutative algebraic groups over a field $k$:
Pass to the associated fppf sheaves and use an injective ...
- 2,102
5
votes
0
answers
126
views
Classification of visible actions for *reducible* representations?
Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
- 2,968
7
votes
3
answers
638
views
Zariski density reference request
I have seen a theorem attributed to Chevalley, to the effect that a sub-semi-group in an algebraic group is Zariski dense if and only if the subgroup it generates is Zariski dense. Would anyone happen ...
- 62.3k
4
votes
0
answers
142
views
Spaces of fixed points
I am reading the paper Space with $\mathbb{G}_{m}$-action, hyperbolic localization and nearby cycles by Timo Richarz and I am having some troubles in understanding the proof of Lemma 1.10.
The setting ...
- 823
8
votes
0
answers
278
views
Has the notion of a unipotent group scheme been studied?
The concept of a unipotent algebraic group over a field has been extensively studied and is fundamental in algebraic geometry. However, has the notion of a unipotent group scheme over a general base ...
- 4,655
2
votes
0
answers
66
views
Irreducibility of Białynicki-Birula cells
Let $X\subset \mathbb{P}^n$ be a smooth complex projective variety, and consider a non-trivial action of $\mathbb{C}^*$ on $X$. For any connected fixed component $Y$ of the fixed point locus, we may ...
2
votes
1
answer
262
views
Prodiscreteness of rational points of algebraic groups
Let $F$ be a field of characteristic 0 complete for a discrete non-archimedean valuation.
Let $G$ be a commutative smooth algebraic group over $F$.
Let us put on $G(F)$ the topology induced by the ...
- 141k
3
votes
0
answers
49
views
Algebraicity of the group of equivariant automorphisms of an almost homogeneous variety
The base field is the field of complex numbers. Let $G$ be a connected linear algebraic group. Let $X$ be an almost homogeneous algebraic variety, i.e. $G$ acts on $X$ with a dense open orbit $U \...
- 543
3
votes
0
answers
117
views
Smooth unipotent algebraic groups over $\mathbb A^n$
Let $G\to \mathbb A^n_{\mathbb C}$ be a smooth morphism whose fibers at any point of $\mathbb A^n$ are unipotent groups. Can we conclude that $G\simeq \mathbb A^{n+N}_{\mathbb C}$ for some $N$, as a ...
- 435
3
votes
0
answers
209
views
Action of an algebraic group $G$ on a scheme $X$ with fixed rational point
Let $X$ be an irreducible locally noetherian $k$-scheme ($k$ any field) and $G$ an algebraic $k$-group acting on $X$.
Proposition 3.1.6 in these notes by M. Brion claims
Let $a : G \times X \to X$ be ...
- 5,780
0
votes
0
answers
113
views
Induced action on infinitesimal thickenings by an algebraic group
Let $X$ be an irreducible locally noetherian $k$-scheme (for $k$ any field), $G$ an algebraic group acting on $X$ via $a:G \times X \to X$ and $x \in X$ a closed point, which is by Zariski's lemma ...
- 5,780
2
votes
1
answer
239
views
Normalizer of Levi subgroup
Let $G$ be a reductive group (we can work on an algebraically closed field if needed) and let $L$ be a parabolic subgroup, i.e. the centralizer of a certain torus $T \subseteq G$.
Associated with this ...
- 576
3
votes
0
answers
150
views
Centers and conjugacy classes of groups relative to a pair of group homomorphisms
$\newcommand{\defeq}{\mathbin{\overset{\mathrm{def}}{=}}}$Given a group $G$, its center $\mathrm{Z}(G)$ and set of conjugacy classes $\mathrm{Cl}(G)$ are defined by
\begin{align*}
\mathrm{Z}(G) &\...
- 11.5k
13
votes
2
answers
2k
views
Is the fixed locus of a group action always a scheme?
Suppose $G$ is an algebraic group with an action $G\times X\to X$ on a scheme. Does the fixed locus (the set of points x∈X fixed by all of $G$) have a scheme structure? You can obviously define the ...
