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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

8 votes
2 answers
2k views

Global Affine Flag Variety and Affine Flag Variety

There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the …
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7 votes
1 answer
710 views

Bialynicki-Birula Decomposition and moment polytopes/graphs

Let $X$ be a possibly singular projective scheme which admits a torus $T$ action and has finitely many $T$ fixed points and one-dimensional $T-$orbits. There are many such schemes in the Grassmannian/ …
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  • 1,719
6 votes
1 answer
498 views

Lattice model for Affine Grassmannians of non type A

There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional $\mathbb{C} …
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  • 1,719
6 votes
0 answers
448 views

Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here. Is there …
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5 votes
1 answer
547 views

Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$

Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel. Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and $S^ …
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5 votes
0 answers
170 views

Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag varie...

This is a follow-up of this previous question below: Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$ Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some …
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  • 1,719
4 votes
0 answers
189 views

Fibers of torus equivariant moment maps

Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map $\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be …
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4 votes
1 answer
821 views

Equi-dimensionality of special fibers in a flat family

Given a flat map $f: X \rightarrow Y$ such that $X$ is a projective variety and $Y$ is a smooth curve. Each generic fiber is isomorphic to an irreducible projective variety $A$ of dimension $d$. The …
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4 votes
5 answers
2k views

Elliptic curves and algebraic stacks

I am a student almost without background on algebraic geometry (but I do know basic graduate algebra and topology). Now I am trying to understand something about algebraic stacks. I want to start wit …
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3 votes
Accepted

Global Affine Flag Variety and Affine Flag Variety

Now let me attempt to give an answer myself. There are very concrete descriptions of the fibers $Fl_{\epsilon}$ in $Fl_{\mathbb{A}^1}$ for each $\epsilon \in \mathbb{A}^1$. $Fl_{\epsilon} \cong LG …
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3 votes
2 answers
686 views

Closure relations between Bruhat cells on the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$. How do we prove the closure relations between the cells, which …
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3 votes
1 answer
236 views

Singularity of torus fixed points from combinatorial data

May I ask what are the relations between the geometry and combinatorics near a torus fixed point? Any references? In particular, let $S$ be a scheme that is torus invariant with finitely many zero and …
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2 votes
2 answers
749 views

Du Val Singularities and Dynkin diagrams references

May I ask whether there are good references for computing blowups of the Du Val Singularities? Also, how are these singularities related to the Dynkin diagrams?
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2 votes
1 answer
485 views

Fibers of the Bott-Samelson Resolution of Schubert Varieties

Is there an explicit (perhaps visual) description of the fibers of the Bott-Samelson Resolutions of Schubert Varieties? Let's fix $G$ to be $GL_n(\mathbb{C})$. Also, how would the answer to the quest …
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2 votes
0 answers
355 views

$G$-equivariant coherent sheaves on Bott$-$Samelson resolutions

Let $G$ be a Lie group and $B$ a Borel subgroup. $G/B$ is the corresponding flag variety. Let $w$ be an element of the Weyl group $W$ with a reduced expression $w = s_1 \cdots s_n$. Let $X_w$ be th …
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