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date: 2022-01-28
tags: [web/quick_notes]
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# 2022-01-28

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## 15:11

- Bott-Samelson manifold: lists of flags starting at the base flag $B/B$ only changing by a little bit at each step.
- It's an iterated $\PP^1$ bundle, hence smooth projective irreducible with a $B$ action.
$\BS^Q$ associated to a word $Q$ in the Weyl group.
- Bott-Samelson map $\BS^Q \to G/B$: send a flag to the top space, its image is some $B\dash$orbit closure $X^w = \cl(BwB)/B$, i.e. a Schubert variety.
Yields that $\BS^Q \to X^w$ is a [resolution of singularities](resolution%20of%20singularities.md), the advantage is that it's combinatorially easy to understand.
  Rational, degree 1.
- What is a [moment map](moment%20map.md)?
- What is a [stratification](stratification)?