# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc: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$f$.
@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.