Ulrich complexity and Picard rank of bidouble Planes
Jerson Caro, Juan Cruz-Penagos, Sergio Troncoso
TL;DR
This work computes the Picard number and Ulrich complexity for general bidouble planes, extending classical results from double planes to non-cyclic abelian covers. It leverages the Galois action to decompose H^2(S,Q) and analyzes the intermediate quotients Y_i to determine when ρ(S)>1, yielding an explicit degree-list criterion. The paper proves nonexistence of Ulrich line bundles in many cases, with explicit low-degree exceptions (0,2,2) and (0,2,4) where Ulrich lines exist, and shows that even bidouble planes typically admit rank-two Ulrich bundles, establishing uc(S,H)=2 in most situations, with refinements for special degree families. The construction of rank-two Ulrich bundles via Cayley–Bacharach–Serre then provides a comprehensive picture of Ulrich complexity for these non-cyclic covers, broadening the scope of Ulrich theory on surfaces of general type.
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 $ρ(S)=1$ 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, and we show that every even bidouble plane carries a special rank-two Ulrich bundle. Our method combines the invariant-theoretic decomposition of $H^2(S,\mathbb{Q})$ under the Galois group with cohomological criteria for Ulrich bundles.