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Abstract
The geometry of Schubert varieties $X(w)$ for a Kac-Moody group $G$ is closely related to the corresponding affine Weyl group $W$. A great deal of geometric information is encoded in the Bruhat order on $W$.In particular, given a pair of elements $x le w$ in $W$, there are integers $q^w_x$ defined using the Bruhat order which can be used to determine rational smoothness of $X(w) $.We prove general results relating the Bruhat order for $W$ of type $tilde A_2$ to the action of $W$ on $mathbb{R}^2, $ using the bijection of $W$ with the center points of the alcoves on $mathbb{R}^2$.We apply these results to an interesting family of elements $w(ell)in W (ell in mathbb{N})$ called spiral elements. We show that $xle w(ell)$ if and only if the corresponding center point $xq$ lies in a region $R(ell)$ which is close to a triangle. Using this we determine all the $q^{w(ell)}_x$ and determine the set ofrationally smooth points of $X(w(ell))$. This leads to the proof of the lookup conjecturefor spiral Schubert varieties $X(w(ell))$.