Hochster θ配对与数值等价
Hochster's theta pairing and numerical equivalence
摘要：
设$(A,\m)$是具有孤立奇点的局部超曲面.我们表明，Hochster的θ配对消失的元素是{数值等价零}的Grothendieck组的$A$的温和的假设下，$\spec A$承认一个奇点的分解。我们还证明了当$\dim A =3$时，Hochster θ配对是半正定的.这些结果表明，对于任何三维孤立超曲面奇异性，如果它不是UFD且具有去奇异化，则存在Dutta-Hochster-McLaughlin关于Serre的交重数一般消失的反例.我们的方法涉及到证明θ给出了态射$\spec A/\m \to \spec A$的一个双变类。如果$A$是具有去奇异化的三维孤立超曲面奇点，则$A$的除子类群是无挠的.
英文摘要：
Let $(A,\m)$ be a local hypersurface with isolated singularity. We show that Hochster's theta pairing vanishes on elements that are {numerically equivalent to zero} in the Grothendieck group of $A$ under the mild assumption that $\spec A$ admits a resolution of singularity. We also prove that when $\dim A =3$, the Hochster's theta pairing is positive semidefinite. These results combine to show that the counter-example of Dutta-Hochster-McLaughlin to general vanishing of Serre's intersection multiplicity exists for any three dimensional isolated hypersurface singularity that is not a UFD and has a desingularization. Our method involves showing that theta gives a bivariant class for the morphism $\spec A/\m \to \spec A$. It also follows that if $A$ is three dimensional isolated hypersurface singularity that has a desingularization, the divisor class group of $A$ is finitely generated torsion-free.
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