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Fibers of generic projections

Part of: Local theory Projective and enumerative geometry Ring extensions and related topics

Published online by Cambridge University Press:  02 February 2010

Roya Beheshti  and
David Eisenbud
Roya Beheshti
Affiliation:
Department of Mathematics, Washington University, St Louis, MO 63130, USA (email: beheshti@math.wustl.edu)
David Eisenbud
Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA (email: eisenbud@math.berkeley.edu)
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Abstract

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Let X be a smooth projective variety of dimension n in Pr, and let π:X→Pn+c be a general linear projection, with c>0. In this paper we bound the scheme-theoretic complexity of the fibers of π. In his famous work on stable mappings, Mather extended the classical results by showing that the number of distinct points in the fiber is bounded by B:=n/c+1, and that, when n is not too large, the degree of the fiber (taking the scheme structure into account) is also bounded by B. A result of Lazarsfeld shows that this fails dramatically for n≫0. We describe a new invariant of the scheme-theoretic fiber that agrees with the degree in many cases and is always bounded by B. We deduce, for example, that if we write a fiber as the disjoint union of schemes Y′ and Y′′ such that Y′ is the union of the locally complete intersection components of Y, then deg Y′+deg Y′′red≤B. Our method also gives a sharp bound on the subvariety of Pr swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ran’s ‘dimension +2 secant lemma’.

Keywords

projective varietylinear projectionsecant lemma

MSC classification

Secondary: 14N05: Projective techniques 14B07: Deformations of singularities 13B22: Integral closure of rings and ideals ; integrally closed rings, related rings (Japanese, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 2 , March 2010 , pp. 435 - 456
Copyright
Copyright © Foundation Compositio Mathematica 2010

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