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

Abstract

LetX 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 xεX, the Jacobian ϱf(x) has rank at least n−2. For r≥1, consider the subscheme N_r of Y defined by the (r−1)th Fitting ideal of the $\mathcal{O}_Y $ -module $f_ * \mathcal{O}_X $ , and set M_r∶=f⁻¹N_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.

If each component of M_r, or equivalently of N_r, has the minimal possible 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 ${}_{f}^{r}$ and the blowup Bl(M_r, M_r+1) is equal to the universal subscheme Univ ${}_{f}^{r}$ of Hilb ${}_{f}^{r}$ ×_YX; moreover, Hilb ${}_{f}^{r}$ and Univ ${}_{f}^{r}$ are Gorenstein. In addition, the structure map h:Hilb ${}_{f}^{r}$ →Y is finite and birational onto its image; and its conductor is equal to the ideal $\mathcal{J}_r $ of N_r+1 in N_r, and is locally self-linked. Reciprocally, $h_ * \mathcal{O}_{Hilb_f^r } $ is equal to $\mathcal{H}om(\mathcal{J}_r ,\mathcal{O}_{N_r } )$ . Moreover, h_*[h⁻¹N_r+1]=(r+1)[N_r+1]. Similar assertions hold for the structure map h₁: Univ ${}_{f}^{r}$ →X if r≥2.