The multiple-point schemes of a finite curvilinear map of codimension one

Let X and Y be smooth varieties of dimensions n−1 and n over an arbitrary algebraically closed field, f :X → Y a finite map that is birational onto its image. Suppose that f is curvilinear; that is, for all x ∈ X, the Jacobian ∂f(x) has rank at least n−2. For r ≥ 1, consider the subscheme Nr of Y defined by the (r − 1)-th Fitting ideal of the OY -module f∗OX , and set Mr := f−1Nr . In this setting—in fact, in a more general setting—we prove the following statements, which show that Mr and Nr behave like reasonable schemes of source and target r-fold points of f . If each component of Mr , or equivalently of Nr , has the minimal possible dimension n − r, then Mr and Nr are Cohen–Macaulay, and their fundamental cycles satisfy the relation, f∗[Mr] = r[Nr ]. Now, suppose that each component of Ms, or of Ns, has dimension n − s for s = 1, . . . , r + 1. Then the blowup Bl(Nr ,Nr+1) is equal to the Hilbert scheme Hilbf , and the blowup Bl(Mr ,Mr+1) is equal to the universal subscheme Univf of Hilb r f ×YX; moreover, Hilbf and Univf are Gorenstein. In addition, the structure map h: Hilbrf → Y is finite and birational onto its image; and its conductor is equal to the ideal Jr of Nr+1 in Nr, and is locally self-linked. Reciprocally, h∗OHilbr f is equal to Hom(Jr ,ONr ). Moreover, h∗[h−1Nr+1] = (r + 1)[Nr+1]. Similar assertions hold for the structure map h1: Univ r f → X if r ≥ 2.