Picard rank and Ulrich line bundles on bidouble planes

Jerson Caro; Juan Cruz-Penagos; Sergio Troncoso

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arXiv · 2512.19544

Picard rank and Ulrich line bundles on bidouble planes

Jerson Caro·Juan Cruz-Penagos·Sergio Troncoso

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Abstract

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 $\rho(S)=1$ 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.

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Jerson Caro, Juan Cruz-Penagos, Sergio Troncoso. 2025-12-22. Picard rank and Ulrich line bundles on bidouble planes. https://arxiv.org/abs/2512.19544

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