Strict Bott–Samelson Resolutions of Schubert Varieties

ABSTRACT

We explore the relationship between two desingularization techniques for Schubert varieties. The Bott–Samelson resolution is the more common of the two, but it fails to encompass many properties that Hironaka resolutions provide, in particular, being an isomorphism over the smooth locus. Using a computer search, a list of cases where Bott–Samelson resolutions having this “strictness” property is compiled for the n = 5, 6 cases. A conjecture based on these results is formulated and is subsequently verified for n = 7, 8. A comparison between Bott–Samelson resolutions and blow-ups is also provided.

KEYWORDS:

• Schubert variety
• Bott-Samelson resolution
• resolution of singularities

2000 AMS SUBJECT CLASSIFICATION:

• 14M15 (Primary)
• 14E15 (Secondary)

Acknowledgments

I would like to thank my advisor Allen Knutson for our many discussions on this topic. This article would not be possible without the ideas that he shared with me.

Funding

This research was supported in part by NSERC PGS-D3 scholarship.