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.

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.

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.

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.

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].

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].

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].

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].

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].

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].