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

LetX andY be smooth varieties of dimensionsn−1 andn over an arbitrary algebraically closed field,f: X→Y a finite map that is birational onto its image. Suppose thatf is curvilinear; that is, for allxεX, the Jacobian ϱf(x) has rank at leastn−2. Forr≥1, consider the subschemeN _r ofY defined by the (r−1)th Fitting ideal of the -module , and setM _r ∶=f ⁻¹ N _r . In this setting—in fact, in a more general setting—we prove the following statements, which show thatM _r andN _r behave like reasonable schemes of source and targetr-fold points off. If each component ofM _r , or equivalently ofN _r , has the minimal possible dimensionn−r, thenM _r andN _r are Cohen-Macaulay, and their fundamental cycles satisfy the relation,f _*[M _r ]=r[N _r ]. Now, suppose that each component ofM _s , or ofN _s , has dimensionn−s fors=1,...,r+1. Then the blowup Bl(N _r ,N _r+1 ) is equal to the Hilbert scheme Hilb _f ^r and the blowup Bl(M _r ,M _r+1 ) is equal to the universal subscheme Univ _f ^r of Hilb _f ^r × _Y X; moreover, Hilb _f ^r and Univ _f ^r are Gorenstein. In addition, the structure maph:Hilb _f ^r →Y is finite and birational onto its image; and its conductor is equal to the ideal ofN _r+1 inN _r , and is locally self-linked. Reciprocally, is equal to . Moreover,h _* [h ⁻¹ N _r+1 ]=(r+1)[N _r+1 ]. Similar assertions hold for the structure maph ₁: Univ _f ^r →X ifr≥2. Supported in part by NSF grant 9106444-DMS. Supported in part by NSA grant MDA904-92-3007, and at MIT 21–30 May 1989 by Sloan Foundation grant 88-10-1. Supported in part by NSF grant DMS-9305832.