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

Steven Kleiman Department of Mathematics, 2-278 MIT Joseph Lipman Department of Mathematics, Purdue University Bernd Ulrich Department of Mathematics, Michigan State University

TBD mathscidoc:1701.332860

Arkiv for Matematik, 34, (2), 285-326, 1995.6
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$xεX$, the Jacobian ϱ$f(x)$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$M$_{$r$}∶=$f$^{−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$f$.
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@inproceedings{steven1995the,
  title={The multiple-point schemes of a finite curvilinear map of codimension one},
  author={Steven Kleiman, Joseph Lipman, and Bernd Ulrich},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203549029984669},
  booktitle={Arkiv for Matematik},
  volume={34},
  number={2},
  pages={285-326},
  year={1995},
}
Steven Kleiman, Joseph Lipman, and Bernd Ulrich. The multiple-point schemes of a finite curvilinear map of codimension one. 1995. Vol. 34. In Arkiv for Matematik. pp.285-326. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203549029984669.
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