Classification of genus-$1$ holomorphic Lefschetz pencils

Noriyuki Hamada; Kenta Hayano

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arXiv · 2008.02925

Classification of genus-$1$ holomorphic Lefschetz pencils

Noriyuki Hamada·Kenta Hayano

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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.

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BibTeX RIS

Noriyuki Hamada, Kenta Hayano. 2020-08-07. Classification of genus-$1$ holomorphic Lefschetz pencils. https://arxiv.org/abs/2008.02925

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