Singular loci of Schubert varieties and the Lookup Conjecture in type $\tilde A_{2}$

Abstract

We describe the loci of non-rationally smooth (nrs) points and of singular points for any non-spiral Schubert variety of $\tilde{A}_2$ in terms of the geometry of the (affine) Weyl group action on the plane $\mathbb{R}^2$. Together with the results of Graham and Li for spiral elements, this allows us to explicitly identify the maximal singular and nrs points in any Schubert variety of type $\tilde{A}_2$. Comparable results are not known for any other infinite-dimensional Kac-Moody flag variety (except for type $\tilde{A}_1$, where every Schubert variety is rationally smooth). As a consequence, we deduce that if $x$ is a point in a non-spiral Schubert variety $X_w$, then $x$ is nrs in $X_w$ if and only if there are more than $\dim X_w$ curves in $X_w$ through $x$ which are stable under the action of a maximal torus, as is true for Schubert varieties in (finite) type $A$. Combined with the work of Graham and Li for spiral Schubert varieties, this implies the Lookup Conjecture for $\tilde{A}_2$.