On Manin's conjecture for quartic del Pezzo fibrations

We generalize the homological sieve method, developed by Das, Lehmann, Tosteson, and the author, to study certain quartic del Pezzo fibrations, and we prove a version of Manin's conjecture over global function fields in these cases. Our proofs combine the $3$-dimensional positive-characteristic minimal model program, the geometry of the space of sections and the Abel--Jacobi mapping, and the homological sieve method. Our quartic del Pezzo surfaces have Picard rank $2$, and admit two birational morphisms to non-split quadric surfaces. In particular, they do not possess any conic fibrations.