All Questions
Tagged with ag.algebraic-geometry algebraic-groups
914
questions
4
votes
0
answers
102
views
Boundary divisor of an equivariant compactification of a non-trivial twist of $\mathbb{G}_a^n$
Let $K$ be a non-perfect field of positive characteristic. Then there exist non-trivial forms of $\mathbb{G}_a^n$ which split over finite purely inseparable extensions of $K$, i.e., algebraic groups $...
2
votes
2
answers
255
views
Zariski closure of the image of an induced representation
Let $G$ be a finitely generated discrete group, $H\le G$ a subgroup of finite index $d$, and let $\rho : H\rightarrow \operatorname{GL}(n,\mathbb{C})$ be a representation.
Let $\tilde{\rho} := \...
- 7,313
0
votes
1
answer
120
views
Functions on products of tori
Let $T$ be an algebraic torus over an algebraically closed field $k$.
Let $d\in\mathbb{N}^{*}$. For every $d$-tuple of integers $\underline{n}=(n_1,\dotsc, n_d)$ and a function $f\in k[T]$, we can ...
- 3,452
2
votes
1
answer
101
views
Rationality of quasi-elementary group actions
I am learning the paper https://arxiv.org/pdf/1604.01005.pdf "Reductive group actions" by Knop and Krotz.
They defined that a linear k-algebraic group $H$ with unipotent radical $H_{u}$ is ...
- 121
2
votes
0
answers
236
views
Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ for $i>0$?
Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
- 1,139
5
votes
1
answer
483
views
Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ for $i>0$?
Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
- 1,139
1
vote
0
answers
107
views
Equivariant cohomology with compact support of the generalized Jacobian of a nodal elliptic curve
[Question forwarded from SE for lack of interaction]
Given an geometric genus $1$ curve with $1$ node $C$ (hence with arithmetic genus $2$), its normalization is given by the elliptic curve $\tilde{C}$...
- 33
1
vote
0
answers
218
views
Reference request: Weil's uniformization theorem
The Weil uniformization theorem says that if $k$ is an algebraically closed field, $G$ a reductive group, and $C$ a curve, we have an isomorphism of stacks $Bun_G(C)\cong G(F_C)\backslash G(\mathbb{A}...
- 1,837
2
votes
0
answers
66
views
On classifying space of normalizer of maximal torus
I am reading Marc Levine's paper 'Motivic Euler characteristics and Witt-valued characteristic classes'. In that paper he considers the $BN_T(SL_n)$, namely the classifying space of the group of ...
- 824
1
vote
0
answers
146
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]^{...
- 767
5
votes
1
answer
206
views
Fields of definition of conjugates
Let $k$ be a field, not necessarily algebraically closed, $G$ an affine group scheme over $k$, $H$ a subgroup of $G$, and $N$ a normal subgroup of $H$, none of them assumed to be smooth. Suppose that ...
- 12k
2
votes
0
answers
165
views
How are tangent spaces related via geometric quotient?
Let $G$ be a linearly reductive group acting regularly on an irreducible affine variety $X$ (over an algebraically closed field of characteristic zero). Suppose there's a $G$-stable open subvariety $U$...
- 767
6
votes
0
answers
269
views
Tits construction of algebraic groups of type D₆ and E₇ via C₃
As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input
an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
- 1,429
2
votes
1
answer
303
views
Product decomposition into semisimple and unipotent parts of an algebraic group (in Borel’s LAG)
Let $G$ be an algebraic group, i.e., an affine reduced, separated $k$-scheme of finite type with structure of a group. In Borel’s Linear Algebraic Groups Theorem III.10.6(4) says
Theorem 10.6 (3): ...
- 5,780
7
votes
3
answers
1k
views
Has anyone researched additive analogues of toric geometry in characteristic zero?
One definition of an $ n $-dimensional toric variety is that it is a variety $ Z $ for which there exists an equivariant embedding of
$ \mathbb{G}_{m}^{n} $ as a Zariski dense, open sub-variety of $ Z ...
- 834