In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities.
It was introduced by Bott & Samelson (1958) in the context of compact Lie groups.
[1] The algebraic formulation is independently due to Hansen (1973) and Demazure (1974).
Let G be a connected reductive complex algebraic group, B a Borel subgroup and T a maximal torus contained in B.
Any such w can be written as a product of reflections by simple roots.
Fix minimal such an expression: so that
ℓ
(ℓ is the length of w.) Let
be the subgroup generated by B and a representative of
be the quotient: with respect to the action of
ℓ
by It is a smooth projective variety.
{\displaystyle X_{w}={\overline {BwB}}/B=(P_{i_{1}}\cdots P_{i_{\ell }})/B}
for the Schubert variety for w, the multiplication map is a resolution of singularities called the Bott–Samelson resolution.
has rational singularities.
[2] There are also some other constructions; see, for example, Vakil (2006).