  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, at every point of X, the
Jacobian has rank at least n-2. For r at least 1, consider the subscheme N_r of
Y defined by the (r-1)st Fitting ideal of the O_Y-module f_*O_X, and set
M_r:=f^{-1}N_r. In this setting --- in fact, in a more general setting --- we
prove the following statements, which show that M_r and N_r behave like
reasonable schemes of source and target r-fold points of f.
  Each component of M_r and N_r is empty or has dimension at least n-r. If each
component of M_r, or equivalently of N_r, has dimension n-r, then M_r and N_r
are Cohen--Macaulay, and their fundamental cycles satisfy the relation,
f_*[M_r]=r[N_r]. Now, suppose that each component of M_s, or of N_s, has
dimension n-s for s=1,...,r+1. Then the blowup Bl(N_r,N_{r+1}) is equal to the
Hilbert scheme Hilb^r_f, and the blowup Bl(M_r,M_{r+1}) is equal to the
universal subscheme Univ^r_f of Hilb^r_f x_Y X; moreover, Hilb^r_f and Univ^r_f
are Gorenstein. In addition, the structure map h:Hilb^r_f->Y is finite and
birational onto its image; and its conductor is equal to the ideal J_r of
N_{r+1} in N_r, and is locally self-linked. Reciprocally, h_*O_{Hilb^r_f} is
equal to Hom(J_r,O_{N_{r}}). Moreover,
  h_*[h^{-1}N_{r+1}]=(r+1)[N_{r+1}].
  Furthermore, similar assertions hold for the structure map h_1:Univ^r_f->X if
r>1.

Algebraic Geometry;Algebraic Geometry