algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, the Bott–Samelson resolution of a Schubert variety is a

resolution of singularities In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, which is a non-singular variety ''W'' with a Proper morphism, proper birational map ''W''→''V''. For varieties ov ...

. It was introduced by in the context of

compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact space, compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are ...

s. The algebraic formulation is independently due to and .

Definition

Let ''G'' be a connected reductive complex

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

, ''B'' a

Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgr ...

and ''T'' a

maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...

contained in ''B''. Let

w \in W = N_G(T)/T.

Any such ''w'' can be written as a product of reflections by simple roots. Fix minimal such an expression: :

\underline = (s_, s_, \ldots, s_)

so that

w = s_ s_ \cdots s_

. (''ℓ'' is the

length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...

of ''w''.) Let

P_ \subset G

be the subgroup generated by ''B'' and a representative of

s_

. Let

Z_

be the quotient: :

Z_ = P_ \times \cdots \times P_/B^\ell

with respect to the action of

B^\ell

by :

(b_1, \ldots, b_\ell) \cdot (p_1, \ldots, p_\ell) = (p_1 b_1^, b_1 p_2 b_2^, \ldots, b_ p_\ell b_\ell^).

It is a smooth

projective variety In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...

. Writing

X_w = \overline / B = (P_ \cdots P_)/B

for the Schubert variety for ''w'', the multiplication map :

\pi: Z_ \to X_w

is a

called the Bott–Samelson resolution.

\pi

has the property:

\pi_* \mathcal_ = \mathcal_

and

R^i \pi_* \mathcal_ = 0, \, i \ge 1.

In other words,

X_w

has rational singularities. There are also some other constructions; see, for example, .

Notes

References

*. *. *. *. *. *. {{DEFAULTSORT:Bott-Samelson resolution Algebraic geometry Singularity theory