# Wintersemester 2018/19: The Grothendieck group of varieties and stacks

Organised by I. Halupczok, B. Klopsch, S. Schröer and M. Zibrowius.

All talks take place Fridays at 12:30 in 25.22.03.73.

## Short description

The Grothendieck group of varieties is the abelian group generated by isomorphism classes {X} of algebraic schemes, modulo the scissor relations {X \ Y} = {X} − {Y} for closed subschemes Y ⊂ X. This mysterious group was first introduced by Grothendieck in a letter to Serre dated August 16, 1964 [Correspondance]. It lies at the heart for motivic arguments, in particular for motivic integration.
In March 2009, the late Ekedahl posted two preprints on arXiv, in which he introduced and applied the Grothendieck group of stacks. Among other things, he proved that the Grothendieck group of stacks is the localization of the Grothendieck group of varieties. This can be applied to finite groups by considering the classifying stack BG. Its class {BG} in the Grothendieck group of stacks yields an algebro-geometric invariant for finite groups G, which is related to rationality problems and invariant theory.

## Schedule

 12.10.18 1.  The Bittner presentation of the Grothendieck group of varieties (Immanuel Halupczok) Main source: [Bittner 2004], Section 2 and 3 Introduce the Grothendieck group of varieties and the Lefschetz class. Explain the Bittner presentation in detail [Bittner 2004]. This relies on resolution of singularities, together with the Weak Factorization Theorem [Wlodarczyk 2003; Abramovich et al. 2003]. Discuss the statement of the latter. 19.10.18 2.  The category of Chow motives and its relation the Grothendieck group of varieties (Peter Arndt) Main source: [Bittner 2004], Section 4. Briefly explain the category of Chow motives following [Scholl 1994], and show that there is a homomorphism of rings from the Grothendieck group of varieties to the Grothendieck group of Chow motives, as explained in [Bittner 2004], Section 4. 26.10.18 3.  The monoid SB of stable birational equivalence classes of varieties (Andre Schell) Main source: [Larsen, Lunts 2003], Section 2 Show that dividing out the Lefschetz class yields the monoid ring on the monoid SB of stable birational equivalence classes of varieties, following Larsen and Lunts [2003], Section 2. 02.11.18 4.  The Grothendieck group of varieties contains zero divisors (Benedikt Schilson) Main source: [Poonen 2002] Show that the Grothendieck group of varieties must contains zero-divisors, according to Poonen [2002]. This relies on some facts about abelian varieties, which should discussed. 09.11.18 5.  Tautological bundle on Hilberst schemes of points (Andreas Krug (Philipps-Universität Marburg)) 16.11.18 6.  Algebraic spaces (Johannes Fischer) Main source: [Olsson 2016; Artin 1971, 1973; Knutson 1971] Discuss the notion of algebraic spaces, a generalization of schemes where the Zariski topology is replaced by the étale topology. Explain the advantages of algebraic space via fundamental examples: Denormalizations of schemes, quotients by finite group actions, contractions of negative-definite curves on surfaces and higher-dimensional generalizations. Sources: for example [Olsson 2016; Artin 1971, 1973; Knutson 1971]. 23.11.18 7.  Stacks with examples (Stefan Schröer) Main source: [Olsson 2016; Fantechi 2001; Laumon and Moret-Bailly 2000] Discuss the notion of algebraic stacks, a further generalization of schemes where the set of A-valued points X(A) from the Yoneda functor is replaced by a fiber category. The objects of these categories are typically schemes or sheaves comprising a moduli problem, but in general are rather unrestricted. Stress examples: the stack of curves of genus g, moduli stacks of principal bundles, and quotient stacks [X/G]. Also discuss inertia stacks. Sources: for example [Olsson 2016; Fantechi 2001; Laumon and Moret-Bailly 2000]. 30.11.18 8.  Explicit resolution of weak wild quotient singularities on arithmetic surfaces (Stefan Wewers (Universität Ulm)) 07.12.18 9.  The Grothendieck group of stacks (Thuong Dang) Main source: [Ekedahl 2009a], Section 1 Introduce the Grothendieck group of stacks, following Ekedahl [2009a], Section 1, and show that the Grothendieck group of stacks is obtained by localizing the Grothendieck group of varieties. See also [Martino 2016, 2017]. 14.12.18 10.  The algebro-geometric invariant {BG} (Leif Zimmermann) Main source: [Ekedahl 2009b], Section 3 and 4 Introduce the algebro-geometric invariant {BG} in the Grothendieck group of stacks, where G is a finite or algebraic group. Discuss the examples of finite groups G with {BG} = 1 given in [Ekedahl 2009b], Section 3 and 4. See also [Martino 2016, 2017]. 21.12.18 No Oberseminar 11.01.19 11.  {BG} and the second group cohomology (Kevin Langlois) Main source: [Ekedahl 2009b], Section 5 Show that there are finite groups G with {BG} non-trivial, and discuss the relation to the second group cohomology, after Ekedahl [2009b], Section 5. Explain the necessary results from Saltman [1984] and Bogomolov [1987] about invariant theory. See also [Martino 2016, 2017]. 18.01.19 12.  Serre’s special group and SO in the Grothendieck group of stacks (Sasa Novakovic) Main source: [Talpo, Vistoli 2017] Discuss Serre’s notion of special group and show that {BG} is the inverse of {G} in the localized Grothendieck group of varieties. Show that the orthogonal groups G = SO also satisfy this relation, following Talpo and Vistoli [2017]. 25.01.19 TBA 01.02.19 Program discussion for the next semester

## Literature

 [Bittner 2004] The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140, 1011–1032. [Ekedahl 2009a] The Grothendieck group of algebraic stacks. Preprint, arXiv:0903.3143. [Ekedahl 2009b] A geometric invariant of a finite group. Preprint, arXiv:0903.3148. [Larsen, Lunts 2003] Motivic measures and stable birational geometry. Mosc. Math. J. 3, 85–95, 259. [Martino 2016] The Ekedahl invariants for finite groups. J. Pure Appl. Algebra 220, 1294–1309. [Martino 2017] Introduction to the Ekedahl invariants. Math. Scand. 120, 211–224. [Poonen 2002] The Grothendieck ring of varieties is not a domain. Math. Res. Lett. 9, 493–497. [Talpo, Vistoli 2017] The motivic class of the classifying stack of the special orthogonal group. Bull. Lond. Math. Soc. 49, 818–823.

Website edited by Sasa Novakovic (2018)