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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2505.11348 |
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| _version_ | 1866909612946489344 |
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| author | Nowroozi, Maryam |
| author_facet | Nowroozi, Maryam |
| contents | Let $K$ be a number field and $S$ a finite set of primes of $K$. Scholl proved that there are only finitely many $K$-isomorphism classes of del Pezzo surfaces of any degree $1 \le d \le 9$ over $K$ with good reduction away from $S$. Let instead $K$ be the cyclotomic $\mathbb{Z}_5$-extension of $\mathbb{Q}$.In this paper, we show, for $d=3$, $4$, that there are infinitely many $\overline{\mathbb{Q}}$ isomorphism classes of del Pezzo surfaces, defined over $K$, with good reduction away from the unique prime above $5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11348 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | del Pezzo surfaces with one bad prime over cyclotomic $\mathbb{Z}_\ell$-extensions Nowroozi, Maryam Number Theory Algebraic Geometry 11G35 (14G05) Let $K$ be a number field and $S$ a finite set of primes of $K$. Scholl proved that there are only finitely many $K$-isomorphism classes of del Pezzo surfaces of any degree $1 \le d \le 9$ over $K$ with good reduction away from $S$. Let instead $K$ be the cyclotomic $\mathbb{Z}_5$-extension of $\mathbb{Q}$.In this paper, we show, for $d=3$, $4$, that there are infinitely many $\overline{\mathbb{Q}}$ isomorphism classes of del Pezzo surfaces, defined over $K$, with good reduction away from the unique prime above $5$. |
| title | del Pezzo surfaces with one bad prime over cyclotomic $\mathbb{Z}_\ell$-extensions |
| topic | Number Theory Algebraic Geometry 11G35 (14G05) |
| url | https://arxiv.org/abs/2505.11348 |