Hochster's theta pairing and numerical equivalence

TL;DR

This paper investigates Hochster's theta pairing on local hypersurfaces with isolated singularities, showing its vanishing on numerically trivial elements and its positive semidefiniteness in dimension three, with implications for Serre's intersection multiplicity.

Contribution

It establishes the vanishing of Hochster's theta pairing on numerically equivalent elements and proves its positive semidefiniteness in three dimensions, linking to singularity resolutions.

Findings

Theta pairing vanishes on numerically trivial elements.

Theta pairing is positive semidefinite in dimension three.

Counterexamples to Serre's intersection multiplicity exist under certain conditions.

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.