Existence of minimal del Pezzo surfaces of degree 1 with conic bundles over finite fields
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917354864115712 |
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| author | Trip, Manoy T. |
| author_facet | Trip, Manoy T. |
| contents | We study minimal del Pezzo surfaces of degree 1 with a conic bundle over a finite field $\mathbb{F}_q$ according to the action of the absolute Galois group on the singular fibers (which is known as their type). We give a lower bound on the size of the field over which they exist, and determine values of $q$ for which certain types cannot exist. In particular, we solve the inverse Galois problem for certain types of minimal del Pezzo surfaces of degree 1 over finite fields with a conic bundle structure. Additionally, we give bounds on the values of $q$ for which del Pezzo surfaces of degree 1 of index 8 exist over $\mathbb{F}_q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20054 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence of minimal del Pezzo surfaces of degree 1 with conic bundles over finite fields Trip, Manoy T. Algebraic Geometry Number Theory We study minimal del Pezzo surfaces of degree 1 with a conic bundle over a finite field $\mathbb{F}_q$ according to the action of the absolute Galois group on the singular fibers (which is known as their type). We give a lower bound on the size of the field over which they exist, and determine values of $q$ for which certain types cannot exist. In particular, we solve the inverse Galois problem for certain types of minimal del Pezzo surfaces of degree 1 over finite fields with a conic bundle structure. Additionally, we give bounds on the values of $q$ for which del Pezzo surfaces of degree 1 of index 8 exist over $\mathbb{F}_q$. |
| title | Existence of minimal del Pezzo surfaces of degree 1 with conic bundles over finite fields |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2603.20054 |