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 ${\mathbb{P}}^1\times {\mathbb{P}}^1$ of bi-degree $(2,2)$ or a blow-up of the pencil on ${\mathbb{P}}^2$ 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 ${\mathbb{P}}^2$ 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 ${\mathbb{P}}^2$ and ${\mathbb{P}}^1\times {\mathbb{P}}^1$ above, in particular these are both holomorphic.

41 pages, many figures

References in corpus (1)

• Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration