Classification of genus-$1$ holomorphic Lefschetz pencils

TL;DR

This paper classifies genus-1 holomorphic Lefschetz pencils up to smooth isomorphism, showing they are either on   or a blow-up of , and analyzes their monodromy and specific examples.

Contribution

It provides a complete classification of genus-1 holomorphic Lefschetz pencils without embedded spheres, including their monodromy factorizations and isomorphism classes.

Findings

Classified genus-1 holomorphic Lefschetz pencils as on   or blow-ups of degree 3 on .

Determined monodromy factorizations for these pencils.

Showed specific constructions by Korkmaz-Ozbagci and Tanaka are isomorphic to the classified models.

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.