# Strict Bott-Samelson Resolutions of Schubert Varieties

## TL;DR

This paper investigates strict Bott-Samelson resolutions of Schubert varieties, compares them with Hironaka resolutions, and verifies conjectures through computational methods for specific cases.

## Contribution

It introduces the concept of strict Bott-Samelson resolutions, formulates a conjecture, and verifies it for cases up to n=8 using computational searches.

## Key findings

- Identified cases where Bott-Samelson resolutions are strict for n=5,6.
- Formulated a conjecture on strictness properties.
- Verified the conjecture for n=7,8.

## 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.