  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.
