## Duke Mathematical Journal

### Discriminants in the Grothendieck ring

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 17 April 2015

https://projecteuclid.org/euclid.dmj/1429282680

Digital Object Identifier
doi:10.1215/00127094-2877184

Mathematical Reviews number (MathSciNet)
MR3336842

Zentralblatt MATH identifier
06443240

Subjects
Primary: 14D06: Fibrations, degenerations

#### Citation

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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