Classification of genus-$1$ holomorphic Lefschetz pencils

Abstract

In this paper, we classify relatively minimal genus-1 holomorphic Lefschetz pencils up to smooth isomorphism. We first show that such a pencil is isomorphic to either the pencil on P 1 × P 1 of bi-degree (2, 2) or a blow-up of the pencil on P 2 of degree 3, provided that no fiber of a pencil contains an embedded sphere. (Note that one can easily classify genus-1 Lefschetz pencils with an embedded sphere in a fiber.) We further determine the monodromy factorizations of these pencils and show that the isomorphism class of a blowup of the pencil on P 2 of degree 3 does not depend on the choice of blown-up base points. We also show that the genus-1 Lefschetz pencils constructed by Korkmaz-Ozbagci (with nine base points) and Tanaka (with eight base points) are respectively isomorphic to the pencils on P 2 and P 1 ×P 1 above, in particular these are both holomorphic.