Stephen Scully: Rational morphisms between quasilinear hypersurfaces

Submission: 2011, Nov 16

We prove analogues of several well-known results concerning rational morphisms 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.

2010 Mathematics Subject Classification: 11E04, 11E76, 14E05, 15A03

Keywords and Phrases: Quasilinear forms, Quasilinear hypersurfaces, Rational morphisms, Birational geometry

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