A regularization theorem for bounded-degree self-maps
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910364328787968 |
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| author | Yang, She |
| author_facet | Yang, She |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07394 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A regularization theorem for bounded-degree self-maps Yang, She Algebraic Geometry Dynamical Systems 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. |
| title | A regularization theorem for bounded-degree self-maps |
| topic | Algebraic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2403.07394 |