Study of local rings with discrete class group and its Picard number

证明了奇素数维Gorenstein孤立商奇点是循环商奇点。我们证明了Hochster定义的theta对给出了Grothendieck群上的一个按数值等价划分的对。由此，我们证明了三维孤立超曲面奇点的类群是无挠的，并且如果它不是UFD，则总存在一个duta-Hochster-MacLauglin型反例。

We proved that a Gorenstein isolated quotient singularity of odd prime dimension is a cyclic quotient singularity. We proved that the theta pairing defined by Hochster gives a pairing over the Grothendieck group divided by numerical equivalence. As a consequence, we proved that the class group of three dimensional isolated hypersurface singularity is torsion-free, and there always exists a counterexample of Dutta-Hochster-MacLauglin type if it is not a UFD.