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Geometriae Dedicata

, Volume 139, Issue 1, pp 331–335 | Cite as

Generically finite morphisms and formal neighborhoods of arcs

  • Lawrence Ein
  • Mircea Mustaţă
Original Paper

Abstract

Let f : XY be a morphism of pure-dimensional schemes of the same dimension, with X smooth. We prove that if \({\gamma\in J_{\infty}(X)}\) is an arc on X having finite order e along the ramification subscheme R f of X, and if its image δ = f (γ) on Y does not lie in J (Y sing), then the induced map T γ J (X) → T δ J (Y) is injective, with a cokernel of dimension e. In particular, if Y is smooth too, and if we denote by \({\widehat{J_{\infty}(X)_{\gamma}}}\) and \({\widehat{J_{\infty}(Y)_{\delta}}}\) the formal neighborhoods of \({\gamma\in J_{\infty}(X)}\) and \({\delta\in J_{\infty}(Y)}\) , then the induced morphism \({\widehat{J_{\infty}(X)_{\gamma}}\to \widehat{J_{\infty}(Y)_{\delta}}}\) is a closed embedding of codimension e.

Keywords

Space of arcs Formal neighborhood Ramification subscheme 

Mathematics Subject Classification (2000)

14B10 14B20 14B25 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

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