Hochster's theta pairing and numerical equivalence

摘要

Let (A,m) be a local hypersurface with an isolated singularity. We show that Hochster's theta pairing theta(A) vanishes on elements that are numerically equivalent to zero in the Grothendieck group of A under the mild assumption that SpecA admits a resolution of singularities. This extends a result by Celikbas-Walker. We also prove that when dimA = 3, Hochster's theta pairing is positive semi-definite. These results combine to show that the counter-example of Dutta-Hochster-McLaughlin to the general vanishing of Serre's intersection multiplicity exists for any three dimensional isolated hypersurface singularity that is not a UFD and has a desingularization. We also show that, if A is three dimensional isolated hypersurface singularity that has a desingularization, the divisor class group is finitely generated torsion-free. Our method involves showing that theta(A) gives a bivariant class for the morphism Spec(A/m) -> SpecA.