Discriminants in the Grothendieck Ring

TL;DR

This paper studies the limiting behavior of discriminants in the Grothendieck ring, describing their stabilization in terms of motivic zeta values and proposing new conjectures on symmetric powers of varieties.

Contribution

It introduces a framework for understanding the stabilization of classes of discriminants in the Grothendieck ring and connects these to motivic zeta values, extending known results in arithmetic and topology.

Findings

Discriminant classes stabilize in the Grothendieck ring as points increase or line bundles become very positive.

Stabilization is described using motivic zeta values.

Conjecture that symmetric powers of irreducible varieties stabilize 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 conjecture that 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 propose a number of new conjectures, both arithmetic and topological.