On the unirationality of conic bundles with discriminant of degree eight
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911278722711552 |
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| author | Casarotti, Alex Gammelgaard, Søren Massarenti, Alex |
| author_facet | Casarotti, Alex Gammelgaard, Søren Massarenti, Alex |
| contents | We study the unirationality of surface conic bundles $π\colon S\to\mathbb P^1$ over an arbitrary field $k$ with discriminant degree $d_S=8$, the first case beyond the del Pezzo range. We divide these surfaces in four families and produce explicit rational multisections via tangent constructions and Cremona transformations. Over $C_1$ fields we obtain Zariski dense loci of minimal, hence non $k$-rational, yet $k$-unirational conic bundles in each family; for one of the types we prove that the dense unirational locus is indeed Zariski open. Finally, we investigate the deformation theory of these conic bundles and how their unirationality behaves under specialization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_17213 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the unirationality of conic bundles with discriminant of degree eight Casarotti, Alex Gammelgaard, Søren Massarenti, Alex Algebraic Geometry Primary 14E08, 14M20, Secondary 14M22, 14J26, 12F20, 12E10 We study the unirationality of surface conic bundles $π\colon S\to\mathbb P^1$ over an arbitrary field $k$ with discriminant degree $d_S=8$, the first case beyond the del Pezzo range. We divide these surfaces in four families and produce explicit rational multisections via tangent constructions and Cremona transformations. Over $C_1$ fields we obtain Zariski dense loci of minimal, hence non $k$-rational, yet $k$-unirational conic bundles in each family; for one of the types we prove that the dense unirational locus is indeed Zariski open. Finally, we investigate the deformation theory of these conic bundles and how their unirationality behaves under specialization. |
| title | On the unirationality of conic bundles with discriminant of degree eight |
| topic | Algebraic Geometry Primary 14E08, 14M20, Secondary 14M22, 14J26, 12F20, 12E10 |
| url | https://arxiv.org/abs/2511.17213 |