Pseudolattices, del Pezzo surfaces, and Lefschetz fibrations

TL;DR

This paper introduces quasi del Pezzo homomorphisms between pseudolattices, classifies them using a pseudolattice minimal model program, and applies these results to classify certain genus one Lefschetz fibrations.

Contribution

It defines quasi del Pezzo homomorphisms, proves their classification, and connects these to the classification of genus one Lefschetz fibrations.

Findings

Classification of quasi del Pezzo homomorphisms established.

Pseudolattice minimal model program developed.

Application to genus one Lefschetz fibrations achieved.

Abstract

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 one Lefschetz fibrations over discs.