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.