All Questions
Tagged with toric-varieties algebraic-groups
11
questions
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
3
votes
1
answer
353
views
When an action on open dense subvariety by an algebraic group extends to variety
A toric variety $X$ over $k$ is a variety which contains an algebraic torus
($T= \mathbb{G}_k^s$)
as a dense open subset such that the action of the torus on itself extends to the whole of
$X$. Slogan:...
- 5,780
5
votes
0
answers
116
views
Where to read about the toric variety coming from a principal nilpotent element of a (semi)simple algebraic group?
Given a principal (regular) nilpotent element $e$ in the Lie algebra $\mathfrak g$ of a complex semisimple algebraic group $G$, let $\mathfrak s=(e,f,h)$ be an $\mathfrak{sl}_2$-triple for $e$. Then ...
- 17.8k
6
votes
2
answers
410
views
Relationship between fans and root data
A (split) reductive linear algebraic group is equivalently described by combinatorial information called a root datum.
A toric variety is described by combinatorial information called a fan.
Both ...
- 3,991
1
vote
0
answers
84
views
How can I compute minimal distance of the AG-code on the Hirzebruch surface $\mathbb{F}_3$?
Let $\mathbb{F}_3$ be the Hirzebruch surface (with index $3$) over a finite field $\mathbb{F}_q$ and $\pi\!: \mathbb{F}_3 \to \mathbb{P}^1$ be the unique $\mathbb{P}^1$-fibration on $\mathbb{F}_3$. ...
- 1,813
7
votes
1
answer
508
views
When are these definitions of "toric variety" equivalent?
Let $k$ be an algebraically closed field. Let $X$ be an integral $k$-scheme, separated and of finite type over $k$. Let $d := \dim X$, let $T := (\mathbb{G}_{m,k})^{d}$ be the $d$-dimensional torus, ...
- 1,997
4
votes
1
answer
332
views
Toric variety defined by the Weyl orbit of a minuscule weight
Let $\Phi$ be a (reduced, crystallographic) root system with Weyl group $\mathcal{W}$, and $p$ a (nonzero) minuscule weight for $\Phi$: its orbit $\mathcal{W}p$ is the set of vertices of a convex ...
- 30.8k
4
votes
1
answer
624
views
Moment map for complete flags variety
Let $M:=U(n)/T^n$ be a complete flag variety, where $U(n)$ is an unitary group and $T^n \simeq (S^1)^n$ consists of its diagonal matrices. I have heard the following construction of a symplectic ...
- 1,990
2
votes
2
answers
629
views
The boundary of toric varieties
Let $\mathcal{X}$ be a toric variety, with $T$ a torus embedded as an open set
in $\mathcal{X}$ (and where the algebraic action of $T$ extends to $\mathcal{X}$). As I am not a toric specialist at all, ...
- 4,503
4
votes
0
answers
313
views
determine if a toric variety is Gorenstein
Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod End(V_{\omega_{i}})\times\prod\...
- 3,452
2
votes
0
answers
565
views
tangent bundle of the toric variety of the wonderful compactification.
Let G be a adjoint group over $k$,algebraically closed of caracteristic zero.
Let $\overline{G}$ be its wonderful compactification.
I denote by $\overline{T}$ the closure of the torus $T$ in $\...
- 3,452