Duke Mathematical Journal

Discriminants in the Grothendieck ring

Ravi Vakil and Melanie Matchett Wood

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We consider the limiting behavior of discriminants, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X and linear systems on X. These are connected—we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motivated by our results, we ask whether the symmetric powers of geometrically irreducible varieties stabilize in the Grothendieck ring (in an appropriate sense). Our results extend parallel results in both arithmetic and topology. We give a number of reasons for considering these questions, and we propose a number of new conjectures, both arithmetic and topological.

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Duke Math. J., Volume 164, Number 6 (2015), 1139-1185.

First available in Project Euclid: 17 April 2015

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Zentralblatt MATH identifier

Primary: 14D06: Fibrations, degenerations

Grothendiek ring stabilization discriminant configuration spaces hypersurfaces Motivic zeta functions


Vakil, Ravi; Wood, Melanie Matchett. Discriminants in the Grothendieck ring. Duke Math. J. 164 (2015), no. 6, 1139--1185. doi:10.1215/00127094-2877184. https://projecteuclid.org/euclid.dmj/1429282680

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