Pseudolattices, del pezzo surfaces, and lefschetz fibrations

Motivated by the relationship between numerical Grothendieck groups induced by the embedding of a smooth anticanonical elliptic curve into a del Pezzo surface, we define the notion of a quasi-del Pezzo homomorphism between pseudolattices and establish its basic properties. The primary aim of the paper is then to prove a classification theorem for quasi-del Pezzo homomorphisms, using a pseudolattice variant of the minimal model program. Finally, this result is applied to the classification of a certain class of genus 1 Lefschetz fibrations over discs.