Invariant hypersurfaces of endomorphisms of the projective 3-space

TL;DR

This paper investigates the structure of hypersurfaces invariant under surjective endomorphisms of degree greater than one on projective 3-space, showing most are hyperplanes with some exceptions and proposing a general conjecture.

Contribution

It proves that invariant hypersurfaces are hyperplanes in projective 3-space, except for four special cases, and conjectures this holds in higher dimensions.

Findings

Invariant hypersurfaces are hyperplanes in most cases

Four exceptional hypersurfaces are identified

Conjecture: all invariant hypersurfaces are hyperplanes in higher dimensions

Abstract

We consider surjective endomorphisms f of degree > 1 on the projective n-space with n = 3, and f^{-1}-stable hypersurfaces V. We show that V is a hyperplane (i.e., deg(V) = 1) but with four possible exceptions; it is conjectured that deg(V) = 1 for any n > 1.