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Rational self-maps of projective surfaces with a regular iterate

Sina Saleh

TL;DR

This paper proves a uniform bound for the time needed for a dominant rational self-map of a complex projective surface to become regular when a regular iterate exists, showing n ≤ 12 (sharp on P^2). It develops a comprehensive framework combining surface classification, valuative trees at infinity, and invariant subvarieties to reduce to toric, P^1-bundle, or P^2/Hirzebruch cases, and then derives extension results for iterates. The results yield a precise dichotomy: either Φ itself is regular, or, under certain non-fibration-preserving conditions, a finite iterate becomes regular with an explicit bound. The work also provides a detailed birational analysis showing that non-elliptic fibrations cannot obstruct regularity, and it furnishes explicit examples illustrating the sharpness of the bounds and constructing long induction chains on toric surfaces.

Abstract

We show that if $Φ: X \dashrightarrow X$ is a dominant rational self-map of a projective surface $X$ over $\mathbb{C}$ with a regular and non-invertible iterate $Φ^n$, then we can take $n \leq 12$. This bound is sharp and realized on $X = \mathbb{P}^2$. In the case where $Φ$ is a birational self-map of $\mathbb{P}^2$ we prove that as long as $Φ$ does not preserve a non-constant fibration, if some iterate $Φ^n$ is regular then $Φ$ itself must be regular.

Rational self-maps of projective surfaces with a regular iterate

TL;DR

This paper proves a uniform bound for the time needed for a dominant rational self-map of a complex projective surface to become regular when a regular iterate exists, showing n ≤ 12 (sharp on P^2). It develops a comprehensive framework combining surface classification, valuative trees at infinity, and invariant subvarieties to reduce to toric, P^1-bundle, or P^2/Hirzebruch cases, and then derives extension results for iterates. The results yield a precise dichotomy: either Φ itself is regular, or, under certain non-fibration-preserving conditions, a finite iterate becomes regular with an explicit bound. The work also provides a detailed birational analysis showing that non-elliptic fibrations cannot obstruct regularity, and it furnishes explicit examples illustrating the sharpness of the bounds and constructing long induction chains on toric surfaces.

Abstract

We show that if is a dominant rational self-map of a projective surface over with a regular and non-invertible iterate , then we can take . This bound is sharp and realized on . In the case where is a birational self-map of we prove that as long as does not preserve a non-constant fibration, if some iterate is regular then itself must be regular.

Paper Structure

This paper contains 25 sections, 39 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Proof strategy for Theorems \ref{['thm:main']} and \ref{['thm:more-precise']}
  3. Proof strategy for Theorem \ref{['thm:birational']}
  4. Notation and terminology.
  5. Outline of the paper.
  6. Examples
  7. Facts about endomorphisms of smooth projective surfaces
  8. Preliminaries and facts about the valuative tree at infinity
  9. Rooted R-trees
  10. Admissible compactifications
  11. Valuations
  12. The tree structure and topology on Vinfty
  13. The main two parametrizations of Vinfty
  14. Induced map of polynomial endomorphisms on Vinfty
  15. Dynamical facts in the case of maximum topological degree
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\Phi: \mathbb P^2 \dashrightarrow \mathbb P^2$ be a dominant rational self-map over $\mathbb C$. Assume that $\Phi^k$ is a non-invertible endomorphism for some $k \ge 1$. Then, $\Phi^i$ is an endomorphism for some $i \in \{1,2,3,4,6, 8, 12\}$. Also, for every $i \in \{1,2,3,4,6, 8, 12\}$ there

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.11
  • ...and 90 more