Discriminants in the Grothendieck Ring

By Ravi Vakil, Melanie Matchett Wood. 2012

72 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/802659b3d7119fef94b19fb1d6c6a5125075ff8c

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In Algebra and Number Theory

In Geometry and Topology

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Citations in the corpus

Correction to the article A spectral sequence for stratified spaces and configuration spaces of points (2016, methodology): “nces for the mod phomology of the unordered conguration spaces: by results of Nagpal [Nag15], our theorem implies that the groups HBM i+2dn(F A(M;n)=Sn;Fp) become eventually periodic. Vakil and Wood [VW15] introduced certain conguration spaces wc (M), depending on a partition . For a suitable choice”
Representation Stability (2014, background): “The representation theory of Sn provides strong constraints on the possible values of dimCH i(Confn(M);C).”