Chow rings of the universal Picard stack in higher genus

Determine the tautological subring and the full Chow ring of the universal degree-d line Picard stack \(\mathrm{Pic}_g^d\) for genus \(g>2\).

Background

The paper expresses universal de Jonquières loci in terms of twisted kappa classes on the universal Picard stack $\mathrm{Pic}_g^d$. In genus zero, the authors explain that the Chow ring can be described explicitly in relevant cases, including a presentation generated by twisted kappa classes when the degree is odd. They then identify the higher-genus situation as unresolved: for $g>2$, neither the tautological subring nor the entire Chow ring of $\mathrm{Pic}_g^d$ is currently known. Determining these rings would clarify the algebraic relations among the tautological classes used in the paper and provide a broader structural foundation for the universal formulas.