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 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.

39 pages, updated with progress by others on various conjectures

References in corpus (10)

• The Grothendieck group of algebraic stacks
• Representation stability in cohomology and asymptotics for families of varieties over finite fields
• Universal Polynomials for Tautological Integrals on Hilbert Schemes
• Universal polynomials for singular curves on surfaces
• Characteristic classes of Hilbert schemes of points via symmetric products
• Some stable homology calculations and Occam's razor for Hodge structures
• Asymptotic behaviour of rational curves
• Homological stability for complements of closures
• Homological stability for coloured configuration spaces and symmetric complements
• Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces

Cited by in corpus (23)

• FI-modules and stability for representations of symmetric groups
• Representation stability in cohomology and asymptotics for families of varieties over finite fields
• A spectral sequence for stratified spaces and configuration spaces of points
• Representation stability for cohomology of configuration spaces in ${\mathbf{R}}^d$
• Semiample Bertini theorems over finite fields
• Topology and arithmetic of resultants, I: spaces of rational maps
• Some stable homology calculations and Occam's razor for Hodge structures
• Computing cohomology of configuration spaces
• Homological stability and densities of generalized configuration spaces
• Coincidences of homological densities, predicted by arithmetic
• $E_n$-cell attachments and a local-to-global principle for homological stability
• A generating function approach to new representation stability phenomena in orbit configuration spaces
• Motivic random variables and representation stability I: Configuration spaces
• The distribution of points on superelliptic curves over finite fields
• Set of independencies and Tutte polynomial of matroids over a domain
• Homological stability for coloured configuration spaces and symmetric complements
• An h-principle for complements of discriminants
• Counting the number of trigonal curves of genus 5 over finite fields
• On the cohomology of certain subspaces of ${Sym}^n({\P }^1)$ and Occam's razor for Hodge structures
• Motivic distribution of rational curves and twisted products of toric varieties
• Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras
• Cohomology of configuration spaces on punctured varieties
• Motivic limits for Fano varieties of $k$-planes