论文标题
Schreieder品种的椭圆形振动和Kodaira尺寸
Elliptic fibrations and Kodaira dimensions of Schreieder's varieties
论文作者
论文摘要
我们讨论了先前由Schreieder构建的某些品种的生育几何形状和Kodaira尺寸，证明在任何维度上，它们都承认椭圆形振动，并且不是一般类型。 $ l $二维品种$ y_n^l $，这是一组$ g \ simeq \ left的$ c_n $ $ c_n $ $ l $ time的商的商（\ simeq \ left（\ sathbb {z}/n \ mathbb {z}数字。如果$ n = 3^c $ schreieder构建了它的明确平滑的模型，而弗拉潘则证明了：这种平滑模型的kodaira尺寸为1，如果$ c> 1 $；如果$ l = 2 $，则是模块化的椭圆表面；如果$ l = 3 $，它将在K3表面上接受振动。 在本文中，我们概括了这些结果：在$ n $和$ l $上没有任何假设，我们证明$ y_n^l $承认许多椭圆纤维及其kodaira尺寸最多是1。如果$ l \ geq 3 $，它具有一个模型模型，该模型可以在K3表面进行振动，并且最多最多为0的Kodaira尺寸（L-1）$ - 尺寸品种的振动。
We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_n^l$, which is the quotient of the product of a certain curve $C_n$ by itself $l$ times by a group $G\simeq\left(\mathbb{Z}/n\mathbb{Z}\right)^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved: the Kodaira dimension of this smooth model is 1, if $c>1$; if $l=2$ it is a modular elliptic surface; if $l=3$ it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on $n$ and $l$ we prove that $Y_n^l$ admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if $l=2$ its minimal resolution is a modular elliptic surface, obtained by a base change of order $n$ on a specific extremal rational elliptic surface; if $l\geq 3$ it has a birational model which admits a fibration in K3 surfaces and a fibration in $(l-1)$-dimensional varieties of Kodaira dimension at most 0.
