Bounded geometry for PCF-special subvarieties
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| author | DeMarco, Laura Mavraki, Niki Myrto Ye, Hexi |
| author_facet | DeMarco, Laura Mavraki, Niki Myrto Ye, Hexi |
| contents | For each integer $d\geq 2$, let $M_d$ denote the moduli space of maps $f: \mathbb{P}^1\to \mathbb{P}^1$ of degree $d$. We study the geometric configurations of subsets of postcritically finite (or PCF) maps in $M_d$. A complex-algebraic subvariety $Y \subset M_d$ is said to be PCF-special if it contains a Zariski-dense set of PCF maps. Here we prove that there are only finitely many positive-dimensional irreducible PCF-special subvarieties in $M_d$ with degree $\leq D$. In addition, there exist constants $N = N(D,d)$ and $B = B(D,d)$ so that for any complex algebraic subvariety $X \subset M_d$ of degree $\leq D$, the Zariski closure $\overline{X\cap\mathrm{PCF}}~$ has at most $N$ irreducible components, each with degree $\leq B$. We also prove generalizations of these results for points with small critical height in $M_d(\bar{\mathbb{Q}})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17343 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounded geometry for PCF-special subvarieties DeMarco, Laura Mavraki, Niki Myrto Ye, Hexi Dynamical Systems Algebraic Geometry Number Theory For each integer $d\geq 2$, let $M_d$ denote the moduli space of maps $f: \mathbb{P}^1\to \mathbb{P}^1$ of degree $d$. We study the geometric configurations of subsets of postcritically finite (or PCF) maps in $M_d$. A complex-algebraic subvariety $Y \subset M_d$ is said to be PCF-special if it contains a Zariski-dense set of PCF maps. Here we prove that there are only finitely many positive-dimensional irreducible PCF-special subvarieties in $M_d$ with degree $\leq D$. In addition, there exist constants $N = N(D,d)$ and $B = B(D,d)$ so that for any complex algebraic subvariety $X \subset M_d$ of degree $\leq D$, the Zariski closure $\overline{X\cap\mathrm{PCF}}~$ has at most $N$ irreducible components, each with degree $\leq B$. We also prove generalizations of these results for points with small critical height in $M_d(\bar{\mathbb{Q}})$. |
| title | Bounded geometry for PCF-special subvarieties |
| topic | Dynamical Systems Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2405.17343 |