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A regularization theorem for bounded-degree self-maps

She Yang

Abstract

Let $K$ be an algebraically closed field of arbitrary characteristic and let $X$ be an irreducible projective variety over $K$. Let $G\subseteq\text{Bir}(X)$ be a bounded-degree subgroup. We prove that there exists an irreducible projective variety $Y$ birational to $X$, such that every element of $G$ becomes an automorphism of $Y$ after the birational transformation. If $K=\mathbb{C}$, this result is stated in [Can14, Theorem 2.5] and the proof backs to [HZ96, Section 5]. The proof in [HZ96] is not purely algebraic. Inheriting the methods in [HZ96], we give a purely algebraic proof of this statement in arbitrary characteristic. We will also discuss a corollary of this result which is useful in arithmetic dynamics.

A regularization theorem for bounded-degree self-maps

Abstract

Let be an algebraically closed field of arbitrary characteristic and let be an irreducible projective variety over . Let be a bounded-degree subgroup. We prove that there exists an irreducible projective variety birational to , such that every element of becomes an automorphism of after the birational transformation. If , this result is stated in [Can14, Theorem 2.5] and the proof backs to [HZ96, Section 5]. The proof in [HZ96] is not purely algebraic. Inheriting the methods in [HZ96], we give a purely algebraic proof of this statement in arbitrary characteristic. We will also discuss a corollary of this result which is useful in arithmetic dynamics.
Paper Structure (16 sections, 47 theorems, 2 equations)

This paper contains 16 sections, 47 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two main tools
  4. Some technical preparations
  5. A baby case of Theorem 1.2
  6. Proof of Theorem 1.2
  7. Step 1
  8. Step 2
  9. Step 3
  10. Step 4
  11. Another approach using Hilbert schemes
  12. Changes in Step 1
  13. Changes in Step 2
  14. Changes in Step 3
  15. Changes in Step 4
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $X$ be an irreducible projective variety and let $G\subseteq\mathrm{Bir}(X)$ be a bounded-degree subgroup. Then there exists an irreducible projective variety $Y$ birational to $X$, such that every element of $G$ becomes an automorphism of $Y$ after the birational transformation.

Theorems & Definitions (55)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 45 more