In this paper, we study the Chazy III,IX and X equations. For the Chazy III equation, by making the birational transformations the Chazy III equation is transformed into a third-order ordinary differential equation of rational type. For this equation, we find its meromorphic solutions, whose free parameters are essentially two. We also show that the system associated with this equation admits new special solutions solved by $tanh(t)$. For the Chazy IX equation, we transform the Chazy IX equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two new birational B{ä}cklund transformations. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we obtain a new partial differential system in two variables involving the Chazy IX equation, This system satisfies the compatibility condition, and admits a travelling wave solution. For the Chazy X equation, we transform the Chazy X equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two birational B{ä}cklund transformations. One of them is new. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we can recover this system.
Bounds for log canonical thresholds with applications to birational rigidity
我们使用相交理论、退化技术和喷流格式来研究对数正则阈值。我们的第一个结果给出了一个下界的对数正则阈值的一对的对数正则阈值的图像通过一个合适的光滑态射。这又是基于与对数正则阈值和Samuel多重性相关的不等式，从而推广了我们之前从数学AG/0205171得到的结果。然后，我们给出了一个由同次齐次方程定义的仿射格式的对数正则阈值的下界，它是关于非对数终端轨迹的维数的（这部分取代了数学AG/0105113）。作为结果的应用，我们证明了P^N中N次光滑超曲面的双有理超刚性，如果4 \leq N \leq 12.
