F-rationality of multiplicity-free subvarieties

A multiplicity-free subvariety of a product of projective lines is a subvariety with the property that the expansion of its Chow class in the usual basis has all coefficients equal to 0 or 1. We show that, over a field of positive characteristic, a multiplicity-free subvariety has F-rational singularities. This strengthens Brion's result that multiplicity-free subvarieties are normal and Cohen--Macaulay, and that they have rational singularities over a field of characteristic 0.