We prove that equivariant multiplicities may be used to determine whether attractive fixed points on -varieties are -smooth, building on the Kumar–Arabia–Brion criterion for rational smoothness. In particular, this gives a combinatorial criterion for the determination of the -smooth locus of Schubert varieties for all primes .
Daniel Juteau. Geordie Williamson. "Kumar’s criterion modulo ." Duke Math. J. 163 (14) 2617 - 2638, 1 November 2014. https://doi.org/10.1215/00127094-2819627