Classification of genus-1holomorphic 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 $P^1\times P^1$ of bidegree $(2,2)$ 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 blow-up 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 $P^1\times P^1$ above, in particular these are both holomorphic.