Picard rank and Ulrich line bundles on bidouble planes

TL;DR

This paper studies the Picard number and Ulrich bundles on bidouble planes, extending classical results to non-cyclic covers and identifying conditions for the existence of Ulrich line bundles.

Contribution

It provides the first systematic analysis of Ulrich bundles on non-cyclic abelian covers, specifically bidouble planes, and determines the Picard number in these cases.

Findings

Picard number is 1 except for specific branch degrees.

Identifies the range of degrees allowing Ulrich line bundles.

Extends classical double plane results to non-cyclic covers.

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.