Classifying sections of del Pezzo fibrations, I

TL;DR

This paper develops a classification strategy for sections of del Pezzo fibrations, proving key lemmas and providing bounds relevant to enumerative geometry and Gromov-Witten invariants.

Contribution

It introduces a novel approach to classify sections of del Pezzo fibrations and proves the Movable Bend and Break lemma in this context.

Findings

Proved the Movable Bend and Break lemma for del Pezzo fibrations

Established upper bounds on the counting function related to sections

Applied results to Gromov-Witten invariants and Abel-Jacobi map studies

Abstract

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.