All Questions
Tagged with ag.algebraic-geometry algebraic-groups
914
questions
2
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174
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Are parabolic Springer fibers equal dimensional?
Let $G$ be a simple algrbraic group （ of type BCDEFG ） over the complex number $\mathbb{C}$, let $P$ be a parabolic subgroup of $G$, suppose we have a resolution of singularities $\mu: T^*(G/P)\to \...
- 551
3
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1
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124
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Connected components of a spherical subgroup from spherical data?
This question is in a similar spirit to this one by Mikhail Borovoi.
Let $G$ be a reductive group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety.
Losev proved that the spherical $...
- 2,386
3
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1
answer
316
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Does a quasi-split reductive group scheme admit a maximal torus?
Let $G \to S$ a reductive group scheme over arbitrary base. Following the conventions from Conrad's Reductive Group Schemes notes, we define a Borel subgroup to be an $S$-subgroup scheme $B \subseteq ...
- 565
3
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1
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193
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What is the minimum possible k-rank of a quasi-split reductive group over a field?
It is not possible for a quasi-split reductive group $G$ over a field $k$ to be anisotropic (unless it is solvable, hence its connected component is a torus). Indeed, there exists a proper $k$-...
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1
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217
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An algebraic stack is an algebraic space if and only if it has the trivial stabilizer group
Let $G\to S$ be a smooth affine group scheme over a scheme. Let $U$ be a scheme over $S$ with an action of $G$. Let $[U/G]$ be the quotient stack.
In Alper's note: Stacks and Moduli, there is a result ...
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3
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1
answer
238
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(non)reduced stabilizer scheme
A well known open question is whether the scheme of commuting pairs in a complex reductive group $G$, for example in $G=GL(n)$, is reduced. The variety of commuting pairs is a special case of a more ...
- 1,526
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164
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Simplicial resolution for commutative group scheme
Let $X$ be a quasi-projective $k$-variety. In this case the symmetric power $S^d(X)$ is well-defined. If $S^\bullet(X)=\bigsqcup_{n>0}S^d(X)$, where we suppose $S^0(X)=\operatorname{spec}(k)$, then ...
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1
answer
339
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Maximal subgroups of projective general linear group
$\newcommand{\sc}{\mathrm{sc}}$All the groups below are algebraic groups over an algebraically closed field,
From Page $163$ of Malle and Testerman's book "Linear algebraic groups and finite ...
- 501
6
votes
1
answer
239
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Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$
This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763.
It got upvotes, but no answers or comments, and so I ask it here.
Let $G$ ...
- 13.8k
6
votes
1
answer
363
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Interpreting group-theoretic sentences as statements about algebraic groups
Suppose we are given a sentence in the language of groups, e.g. $\phi=\forall x\forall y(x\cdot y=y\cdot x)$, and suppose that we are also given the data defining an algebraic group $G/k$. One can ...
- 333
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111
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Krull dimension of ring of invariants
Let $A$ be a $K$-algebra for some local number field $K$, and denote by $\dim A$ its Krull dimension. Let $G$ be an algebraic group defined over $\text{Spec}K$, and assume $G$ acts on $A$ by $K$-...
- 2,113
1
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154
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Centraliser of a finite group
Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$.
We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...
- 501
1
vote
1
answer
269
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Subgroups of $\operatorname{PGL}_n$
As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, $\operatorname{GO}_{2n}$ is a closed normal subgroup of the conformal orthogonal group $\operatorname{CO}_{2n}$. ...
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3
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102
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When can we lift transitivity of an action from geometric points to a flat cover?
Let $G$ a nice group scheme (say, over $S$), $X$ a smooth $G$-scheme over $S$, that is, $\pi : X \to S$ a smooth, $G$-invariant morphism. Assume that the action is transitive on algebraically closed ...
- 565
2
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1
answer
262
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Commutative group scheme cohomology on generic point
Setup:
Let $k$ be an algebraically closed field.
Let $C$ be a smooth connected projective curve over $k$.
Let $J$ be a smooth commutative group scheme over $C$ with connected fibers.
Let $j:\eta\to C$ ...
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