Picard rank and Ulrich line bundles on bidouble planes
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908824573575168 |
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| author | Caro, Jerson Cruz-Penagos, Juan Troncoso, Sergio |
| author_facet | Caro, Jerson Cruz-Penagos, Juan Troncoso, Sergio |
| contents | We determine the Picard number and the Ulrich complexity of general bidouble covers of the projective plane, providing the first systematic study of Ulrich bundles on non-cyclic abelian covers. For a bidouble plane branched along three smooth curves of degrees $n_1,n_2,n_3$, we show that $ρ(S)=1$ unless $(n_1,n_2,n_3)$ belongs to an explicit list, thereby extending Buium's classical results on double planes to the non-cyclic case. As an application, we determine the range of branch degrees for which Ulrich line bundles could exist. Our method combines the invariant-theoretic decomposition of $H^2(S,\mathbb{Q})$ under the Galois group with cohomological criteria for Ulrich bundles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Picard rank and Ulrich line bundles on bidouble planes Caro, Jerson Cruz-Penagos, Juan Troncoso, Sergio Algebraic Geometry Number Theory 14C22, 14C20, 14E20, 14J60 We determine the Picard number and the Ulrich complexity of general bidouble covers of the projective plane, providing the first systematic study of Ulrich bundles on non-cyclic abelian covers. For a bidouble plane branched along three smooth curves of degrees $n_1,n_2,n_3$, we show that $ρ(S)=1$ unless $(n_1,n_2,n_3)$ belongs to an explicit list, thereby extending Buium's classical results on double planes to the non-cyclic case. As an application, we determine the range of branch degrees for which Ulrich line bundles could exist. Our method combines the invariant-theoretic decomposition of $H^2(S,\mathbb{Q})$ under the Galois group with cohomological criteria for Ulrich bundles. |
| title | Picard rank and Ulrich line bundles on bidouble planes |
| topic | Algebraic Geometry Number Theory 14C22, 14C20, 14E20, 14J60 |
| url | https://arxiv.org/abs/2512.19544 |