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

Abstract

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.

minor revision, 36 pages, to appear in Simons Symposia "Geometry over non-closed fields"