Totally Invariant Divisors of non Trivial Endomorphisms of the Projective Space

TL;DR

This paper investigates totally invariant divisors under non-isomorphic endomorphisms of complex projective space, establishing degree bounds and proving their linearity when singularities are isolated.

Contribution

It provides an upper bound for the degree of totally invariant divisors and proves their linearity under certain singularity conditions, advancing understanding of their structure.

Findings

Upper bound for the degree of totally invariant divisors

Proof of linearity for divisors with isolated singularities

Supports the conjecture that such divisors are unions of hyperplanes

Abstract

It is expected that a totally invariant divisor of a non-isomorphic endomorphism of the complex projective space is a union of hyperplanes. In this paper, we compute an upper bound for the degree of such a divisor. As a consequence, we prove the linearity of totally invariant divisors with isolated singularities.