Local-global principle and integral Tate conjecture for certain varieties
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916409932513280 |
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| author | Tian, Zhiyu |
| author_facet | Tian, Zhiyu |
| contents | We give a geometric criterion to check the validity of the integral Tate conjecture for one-cycles on a smooth projective variety that is separably rationally connected in codimension one, and to check that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on a separably rationally connected variety defined over a global function field.
We prove that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on all geometrically rational surfaces defined over a global function field, and to the Hasse principle for rational points on del Pezzo surfaces of degree four defined over a global function field of odd characteristic.
Along the way, we also prove some results about the space of one-cycles on a smooth projective variety that is separably rationally connected in codimension one, which leads to the equality of the coniveau filtration and the strong coniveau filtration on degree $3$ homology of such varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_15915 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Local-global principle and integral Tate conjecture for certain varieties Tian, Zhiyu Algebraic Geometry 14M22, 14G12, 14G25, 14C25 We give a geometric criterion to check the validity of the integral Tate conjecture for one-cycles on a smooth projective variety that is separably rationally connected in codimension one, and to check that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on a separably rationally connected variety defined over a global function field. We prove that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on all geometrically rational surfaces defined over a global function field, and to the Hasse principle for rational points on del Pezzo surfaces of degree four defined over a global function field of odd characteristic. Along the way, we also prove some results about the space of one-cycles on a smooth projective variety that is separably rationally connected in codimension one, which leads to the equality of the coniveau filtration and the strong coniveau filtration on degree $3$ homology of such varieties. |
| title | Local-global principle and integral Tate conjecture for certain varieties |
| topic | Algebraic Geometry 14M22, 14G12, 14G25, 14C25 |
| url | https://arxiv.org/abs/2211.15915 |