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Graph Hypersurfaces and a Dichotomy in the Grothendieck Ring

Aluffi, Paolo and Marcolli, Matilde (2011) Graph Hypersurfaces and a Dichotomy in the Grothendieck Ring. Letters in Mathematical Physics, 95 (3). pp. 223-232. ISSN 0377-9017.

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The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of Z generated by the class of a point. This clarifies the extent to which the graph hypersurfaces ‘generate the Grothendieck ring of varieties’: while it is known that graph hypersurfaces generate the Grothendieck ring over a localization of Z[L] in which L becomes invertible, the span of the graph hypersurfaces in the Grothendieck ring itself is nearly killed by setting the Lefschetz motive L to zero. In particular, this shows that the graph hypersurfaces do not generate the Grothendieck ring prior to localization. The same result yields some information on the mixed Hodge structures of graph hypersurfaces, in the form of a constraint on the terms in their Deligne–Hodge polynomials. These observations are certainly not surprising for the expert reader, but are somewhat hidden in the literature. The treatment in this note is straightforward and self-contained.

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Additional Information:© 2011 Springer. Received: 1 June 2010; Revised: 14 December 2010; Accepted: 12 January 2011; Published online: 29 January 2011.
Subject Keywords:graph hypersurfaces, Grothendieck ring, stable birational equivalence
Issue or Number:3
Classification Code:Mathematics Subject Classification (2000): 81T15, 14E08, 16E20
Record Number:CaltechAUTHORS:20110318-142126748
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:22977
Deposited By: Tony Diaz
Deposited On:18 Mar 2011 21:46
Last Modified:03 Oct 2019 02:42

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