Singular loci of Schubert varieties and the Lookup Conjecture in type A₂

We describe the loci of non-rationally smooth (nrs) points and of singular points for any non-spiral Schubert variety of Ã₂ in terms of the geometry of the (affine) Weyl group action on the plane R². 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 Ã₂. Comparable results are not known for any other infinite-dimensional Kac-Moody flag variety (except for type Ã₁, where every Schubert variety is rationally smooth). As a consequence, we deduce that if x is a point in a non-spiral Schubert variety Xw, then x is nrs in Xw if and only if there are more than Xw curves in Xw 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 Ã₂.