On the variety of Borels in relative position $\vec{w}$

Let $G$ be a connected semi-simple group defined over and algebraically closed field, $T$ a fixed Cartan, $B$ a fixed Borel containing $T$, $S$ a set of simple reflections associated to the simple positive roots corresponding to $(T,B)$, and let $\mathcal{B}\cong G\mathrm{/}B$ denote the Borel variety. For any $s_i\in S$,$1\leq i\leq n$,let$\bar{O}(s_1,…, s_n)= {(B_0,…, B_{n})\in {\cal B}^{n+1} | (B_{i-1},B_{i})\in \bar{O(s_i)}, 1\leq i\leq n}$,where$O(s)$denotesthesubvarietyofpairsofBorelsin${\cal B}^2$inrelativeposition$s$. We show that such varieties are smooth and indicate why this result is, in one sense, best possible. Our main results assume that $k$ has characteristic 0.