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Rational maps between quasilinear hypersurfaces

Part of: Birational geometry Forms and linear algebraic groups Basic linear algebra

Published online by Cambridge University Press:  17 December 2012

Stephen Scully
Stephen Scully*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK (email: [email protected])
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Abstract

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We prove analogues of several well-known results concerning rational maps between quadrics for the class of so-called quasilinear p-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alternative approach, which is partly facilitated by the appearance of several non-traditional features in the study of these objects from an algebraic perspective. Our main results were previously known for the class of quasilinear quadrics. We provide new proofs here, because the original proofs do not immediately generalise for quasilinear hypersurfaces of higher degree.

Keywords

quasilinear formsquasilinear hypersurfacesrational mapsbirational geometry

MSC classification

Secondary: 11E04: Quadratic forms over general fields 11E76: Forms of degree higher than two 14E05: Rational and birational maps 15A03: Vector spaces, linear dependence, rank
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 3 , March 2013 , pp. 333 - 355
Copyright
Copyright © 2012 The Author(s)

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