Nash multiplicity sequences and Hironaka's order function

When X is a d-dimensional variety deﬁned over a ﬁeld k of characteristic zero, a constructive resolution of singularities can be achieved by successively lowering the maximum multiplicity via blow ups at smooth equimultiple centers. This is done by stratifying the maximum multiplicity locus of X by means of the so called resolution functions. The most important of these functions is what we know as Hironaka"s order function in dimension d. Actually, this function can be deﬁned for varieties when the base ﬁeld is perfect; however if the characteristic of k is positive, the function is, in general, too coarse and does not provide enough information so as to deﬁne a resolution. It is very natural to ask what the meaning of this function is in this case, and to try to ﬁnd reﬁnements that could lead, ultimately, to a resolution. In this paper we show that Hironaka"s order function in dimension d can be read in terms of the Nash multiplicity sequences introduced by Lejeune-Jalabert. Therefore, the function is intrinsic to the variety and has a geometrical meaning in terms of its space of arcs.

Nash multiplicity sequences and Hironaka's order function Articles