Discriminants in the Grothendieck ring

Abstract

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.

Homological stability for topological chiral homology of completions

“…This simplifies if M is formal, giving for example the following result in the case M = CP 2 . A minimal model for F 1 (CP 2 , Sym c (S n ) Q ) is given by the CDGA with algebra the free graded-commutative algebra generated by This recovers known results, in particular the case c → ∞ coming from the Dold-Thom theorem and the case c = 1 discussed in [KM14a], which computes the stable homology of w 1 j (CP 2 ) = C j (CP 2 ) to answer Conjecture G of [VW15].…”

“…+ a n ) of j + k obtained by adding j ones to λ. After observing that a similar stability result holds in the Grothendieck ring of varieties (Theorem 1.30a of [VW15]), Vakil and Wood made the following conjecture (Conjecture F of [VW15]): Conjecture 1.3 (Vakil-Wood). For all partitions λ and irreducible smooth complex varieties X, the groups H i (W 1 j λ (X); Q) are independent of j for j sufficiently large compared to i.…”

Representation stability in cohomology and asymptotics for families of varieties over finite fields

“…We remark that the results of [KM] violate a tentative prediction of the stable Betti numbers made in [VW,Eq. 1.50]; for example, for d = 1 this "motivic Occam's Razor" predicted that dim H i (Conf ′ (C); Q) = 2 when i ≡ 0, 1 mod 4, rather than for all i > 0.…”

Motivic random variables and representation stability II: Hypersurface sections

“…For an orientable manifold M , cohomological stabilization for the sequence of spaces Conf τ¨˚n´|τ | M was observed as a consequence of representation stability already by Church [4,Theorem 5]. The existence of the limit appearing in Corollary B for an algebraic variety Y was also shown by Vakil-Wood [14]. Thus, our main contribution is the explicit formula in terms of the M k prY s HS q.…”

“…In this paper, we bring together two recent threads in the theory of cohomological stability for configuration spaces: the theory of representation stability of Church [4] and Church-Farb [7] and the motivic approach of Vakil-Wood [14]. We show that the families of local systems on unordered configuration spaces of algebraic varieties studied in representation stability have natural motivic avatars in the Grothendieck ring of varieties, and that these stabilize under suitable motivic measures (cf.…”