论文标题
关于理性双点喷气方案的奇异纤维的配置
On the configuration of the singular fibers of jet schemes of rational double points
论文作者
论文摘要
对于每个品种$ x $和一个非负整数$ m $，在$ x $上有一个$ x_m $ x_m $，称为$ x $ $ m $ $ x $的JET计划，参数化$ m $ m $ th Jets $ x $。它在$ x $的单数点上的纤维称为单数纤维。对于具有理性双点的表面，穆尔塔达（Mourtada）在$ x_m $的奇异纤维的不可还原组件与最低分辨率$ x $ for $ m \ gg 0 $之间的出色曲线之间给出了一对一的对应关系。 在本文中，对于$ a_n $或$ d_4 $ type的奇异性的表面$ x $，我们研究了奇异纤维的不可约组件的相互作用，并使用此信息构建图。该图的顶点对应于单数纤维的不可约成分，当相应组件的相交对于包容性关系最大时，将连接两个顶点。对于$ a_n $或$ d_4 $ -type奇异性，我们表明该图是$ m \ gg 0 $的分辨率图的同构。
To each variety $X$ and a nonnegative integer $m$, there is a space $X_m$ over $X$, called the jet scheme of $X$ of order $m$, parametrizing $m$-th jets on $X$. Its fiber over a singular point of $X$ is called a singular fiber. For a surface with a rational double point, Mourtada gave a one-to-one correspondence between the irreducible components of the singular fiber of $X_m$ and the exceptional curves of the minimal resolution of $X$ for $m \gg 0$. In this paper, for a surface $X$ over complex number with a singularity of $A_n$ or $D_4$-type, we study the intersections of irreducible components of the singular fiber and construct a graph using this information. The vertices of the graph correspond to irreducible components of the singular fiber and two vertices are connected when the intersection of the corresponding components is maximal for the inclusion relation. In the case of $A_n$ or $D_4$-type singularity, we show that this graph is isomorphic to the resolution graph for $m \gg 0$.
