Arkiv för Matematik

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  • Volume 34, Number 2 (1996), 285-326.

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

Steven Kleiman, Joseph Lipman, and Bernd Ulrich

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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 Nr of Y defined by the (r−1)th Fitting ideal of the $\mathcal{O}_Y $ -module $f_ * \mathcal{O}_X $ , 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 Hilb ${}_{f}^{r}$ and the blowup Bl(Mr, Mr+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 Nr+1 in Nr, 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−1Nr+1]=(r+1)[Nr+1]. Similar assertions hold for the structure map h1: Univ ${}_{f}^{r}$ →X if r≥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.

Article information

Ark. Mat., Volume 34, Number 2 (1996), 285-326.

Received: 16 June 1995
First available in Project Euclid: 31 January 2017

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1996 © Institut Mittag-Leffler


Kleiman, Steven; Lipman, Joseph; Ulrich, Bernd. The multiple-point schemes of a finite curvilinear map of codimension one. Ark. Mat. 34 (1996), no. 2, 285--326. doi:10.1007/BF02559549.

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