Zero-cycles on quasi-projective surfaces over $p$-adic fields
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917535050366976 |
|---|---|
| author | Gazaki, Evangelia Rathore, Jitendra |
| author_facet | Gazaki, Evangelia Rathore, Jitendra |
| contents | A conjecture of Colliot-Thélène predicts that for a smooth projective variety $X$ over a finite extension $k$ of $\mathbb{Q}_p$ the kernel of the Albanese map $\text{CH}_0(X)^{\text{deg}=0}\to Alb_X(k)$ is the direct sum of a divisible group and a finite group. In this article we show that if $π:X\dashrightarrow Y$ is a generically finite rational map between smooth projective surfaces and the conjecture is true for $X\otimes_k L$ for every finite extension $L/k$, then it is true for $Y$. Using work of Raskind and Spiess, this proves the conjecture for surfaces that are geometrically dominated by products of curves, under some assumptions on the reduction type of the Jacobians. The method involves studying similar questions for an open subvariety $U$ of a projective surface $X$ by replacing the Chow group of $0$-cycles with Suslin's singular homology $H_0^{\text{sus}}(U)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20076 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zero-cycles on quasi-projective surfaces over $p$-adic fields Gazaki, Evangelia Rathore, Jitendra Algebraic Geometry A conjecture of Colliot-Thélène predicts that for a smooth projective variety $X$ over a finite extension $k$ of $\mathbb{Q}_p$ the kernel of the Albanese map $\text{CH}_0(X)^{\text{deg}=0}\to Alb_X(k)$ is the direct sum of a divisible group and a finite group. In this article we show that if $π:X\dashrightarrow Y$ is a generically finite rational map between smooth projective surfaces and the conjecture is true for $X\otimes_k L$ for every finite extension $L/k$, then it is true for $Y$. Using work of Raskind and Spiess, this proves the conjecture for surfaces that are geometrically dominated by products of curves, under some assumptions on the reduction type of the Jacobians. The method involves studying similar questions for an open subvariety $U$ of a projective surface $X$ by replacing the Chow group of $0$-cycles with Suslin's singular homology $H_0^{\text{sus}}(U)$. |
| title | Zero-cycles on quasi-projective surfaces over $p$-adic fields |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2507.20076 |