Hochster's theta pairing and numerical equivalence

Abstract

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.

References in corpus (2)

• Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free
• Hochster's Theta Pairing and Algebraic Equivalence