论文标题
属的分类 - $ 1 $ HOLOMORTHIC LEFSCHETZ铅笔
Classification of genus-$1$ holomorphic Lefschetz pencils
论文作者
论文摘要
在本文中，我们将相对最小的属分类为$ 1 $ holomorphic lefschetz铅笔，以使同构平滑。我们首先表明，这种铅笔是同构的，这是$ \ mathbb {p}^1 \ times \ times \ mathbb {p}^1 $ bi-degree $（2,2）$的铅笔，或者在不包含一份pord的pord conder的pord comped and pulber的$ 3 $ 3 $ 3 $ $ 3 $ $ 3 $ 3 $的$ \ Mathbb {p}^2 $ 3 $上。 （请注意，一个人可以轻松地对属 - $ 1 $ lefschetz铅笔进行分类。我们还表明，由korkmaz-ozbagci（带有九个基点）和田中（有8个基点）构建的属属 - $ 1 $ lefschetz铅笔在$ \ mathbb {p}^2 $ and $ \ mathbb {p} p} p} pp {p}^1 $ \ bb {p}^1 pp^1 $ {p} pp^1 $ {Pens^1 $ {p} pp^1 $ {pp^1 $ {全态。
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.
