Rational curves on del Pezzo surfaces in positive characteristic

We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic 0. We also investigate the principles of Geometric Manin's Conjecture for weak del Pezzo surfaces. In the course of this investigation, we give examples of weak del Pezzo surfaces defined over $\mathbb{F}_{2}(t)$ or $\mathbb{F}_{3}(t)$ such that the exceptional sets in Manin's Conjecture are Zariski dense.