The spaces of rational curves on del Pezzo surfaces via conic bundles

Using the homological sieve method developed by Das–Lehmann–Tosteson and the author, we prove Peyre’s all height approach to Manin’s conjecture for split quintic del Pezzo surfaces defined over ${\mathbb{F}}_q(t)$ assuming $q$ is sufficiently large. We also establish lower bounds of correct magnitude for the counting function of rational curves on split low degree del Pezzo surfaces defined over ${\mathbb{F}}_q$ assuming $q$ is large.