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Hochster's theta pairing and numerical equivalence

Published online by Cambridge University Press:  28 July 2014

Hailong Dao  and
Kazuhiko Kurano
Hailong Dao
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523, USAhdao@ku.edu
Kazuhiko Kurano
Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, Higashimata 1-1-1, Tama-ku, Kawasaki-shi 214-8571, Japankurano@isc.meiji.ac.jp
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Abstract

Let (A, ) be a local hypersurface with an isolated singularity. We show that Hochster's theta pairing θA 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 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 θA gives a bivariant class for the morphism Spec (A/) → Spec A.

Keywords

local hypersurfaceisolated singularityHochster's theta invariantintersection multiplicitydivisor class groupGrothendieck groupnumerical equivalences13D0713D1513D2214C1714C35
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 3 , December 2014 , pp. 495 - 525
Copyright
Copyright © ISOPP 2014 

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