Totally invariant divisors of endomorphisms of projective spaces

TL;DR

This paper investigates the structure of totally invariant divisors under endomorphisms of projective spaces, proving linearity in certain cases and establishing bounds on their non-normal loci using logarithmic differentials.

Contribution

It introduces a new approach using logarithmic differentials to analyze invariant divisors and proves their linearity in the case of -dimensional projective space.

Findings

Established a lower bound for the degree of the non-normal locus

Proved that totally invariant divisors are unions of linear spaces in -dimensional projective space

Extended understanding of the structure of invariant divisors in algebraic geometry

Abstract

Totally invariant divisors of endomorphisms of the projective space are expected to be always unions of linear spaces. Using logarithmic differentials we establish a lower bound for the degree of the non-normal locus of a totally invariant divisor. As a consequence we prove the linearity of totally invariant divisors for $\mathbb P^3$.