Rational points on 3-folds with nef anti-canonical class over finite fields
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866915133538697216 |
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| author | Bernasconi, Fabio Filipazzi, Stefano |
| author_facet | Bernasconi, Fabio Filipazzi, Stefano |
| contents | We prove that a geometrically integral smooth 3-fold $X$ with nef anti-canonical class and negative Kodaira dimension over a finite field $\mathbb{F}_q$ of characteristic $p>5$ and cardinality $q=p^e > 19$ has a rational point. Additionally, under the same assumptions on $p$ and $q$, we show that a 3-fold $X$ with trivial canonical class and non-zero first Betti number $b_1(X) \neq 0$ has a rational point. Our techniques rely on the Minimal Model Program to establish several structure results for generalized log Calabi--Yau 3-fold pairs over perfect fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10824 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rational points on 3-folds with nef anti-canonical class over finite fields Bernasconi, Fabio Filipazzi, Stefano Algebraic Geometry We prove that a geometrically integral smooth 3-fold $X$ with nef anti-canonical class and negative Kodaira dimension over a finite field $\mathbb{F}_q$ of characteristic $p>5$ and cardinality $q=p^e > 19$ has a rational point. Additionally, under the same assumptions on $p$ and $q$, we show that a 3-fold $X$ with trivial canonical class and non-zero first Betti number $b_1(X) \neq 0$ has a rational point. Our techniques rely on the Minimal Model Program to establish several structure results for generalized log Calabi--Yau 3-fold pairs over perfect fields. |
| title | Rational points on 3-folds with nef anti-canonical class over finite fields |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2308.10824 |