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Compositio Mathematica Compositio Mathematica

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Some Properties of Finite Morphisms on Double Points

Published online by Cambridge University Press:  04 December 2007

Hassan Haghighi ,
Joel Roberts  and
Rahim Zaare-Nahandi
Hassan Haghighi
Affiliation:
Department of Mathematics, University of Tehran, Iran
Joel Roberts
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA. E-mail: roberts@math.umn.edu
Rahim Zaare-Nahandi
Affiliation:
University of Tehran, and Cent. Theo. Phys. Math., P.O.Box 11365-8486, Tehran, Iran. E-mail: rahimzn@khayam.ut.ac.ir
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Abstract

For a finite morphism f : X → Y of smooth varieties such that f maps X birationally onto X′=f(X), the local equations of f are obtained at the double points which are not triple. If $\cal C$ is the conductor of X over X′, and $D = Sing(X') ⊂ X'$, $Δ ⊂ X$ are the subschemes defined by $\cal C$, then D and Δ are shown to be complete intersections at these points, provided that $\cal C$ has “the expected” codimension. This leads one to determine the depth of local rings of X′ at these double points. On the other hand, when $\cal C$ is reduced in X, it is proved that X′ is weakly normal at these points, and some global results are given. For the case of affine spaces, the local equations of X′ at these points are computed.

Keywords

generic projections double points singular locus conductor complete intersection weak normality seminormality local equations
Type
Research Article
Information
Compositio Mathematica , Volume 121 , Issue 1 , March 2000 , pp. 35 - 53
Copyright
© 2000 Kluwer Academic Publishers

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