Resolving Toric Varieties with Nash Blowups

Let X be a variety over an algebraically closed field K . Its Nash blow-up is a variety over K with a projective morphism to X , which is an isomorphism over the smooth locus. Roughly speaking, it parametrizes all limits of tangent planes to X (a precise definition is given in §2 below). The Nash blow-up of a singular X is not always smooth but seems, in some sense, to be less singular than X . Strictly speaking this is false, for in characteristic p > 0, as explained by Nobile [14], the plane curve x − y = 0 is its own Nash blow-up for any q > 0. In this and other ways the ordinary Nash blow-up proves intractable. However, let the normalized Nash blow-up be the normalization of the Nash blow-up. Then, of course, the normalized Nash blow-up of every curve is smooth. The normalized Nash blow-up of a surface can be singular, but Hironaka [10] and Spivakovsky [7, 16] have shown that every surface becomes smooth after finitely many normalized Nash blow-ups. Thus we are drawn to ask the following.