Pseudolattices, del Pezzo surfaces, and Lefschetz fibrations

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.

v2: Minor revisions at the request of the referee. Remark 3.24 is new, as is the discussion of the noncommutative setting in the final section