Rational morphisms between quasilinear hypersurfaces

TL;DR

This paper extends known results about rational morphisms from quadrics to a broader class of quasilinear p-hypersurfaces, overcoming geometric challenges with algebraic methods.

Contribution

It generalizes rational morphism results from quadrics to higher-degree quasilinear hypersurfaces using novel algebraic approaches.

Findings

Analogues of classical results for quadrics are established for quasilinear p-hypersurfaces.

New proofs are provided that apply to higher-degree hypersurfaces.

The study reveals algebraic features crucial for understanding these hypersurfaces.

Abstract

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.