Classifying sections of del Pezzo fibrations, I

We develop a strategy to classify the components of the space of sections of a del Pezzo fibration over ${\mathbb{P}}^1$. In particular, we prove the Movable Bend and Break lemma for del Pezzo fibrations. Our approach is motivated by Geometric Manin’s Conjecture and proves upper bounds on the associated counting function. We also give applications to enumerativity of Gromov-Witten invariants and to the study of the Abel-Jacobi map.