The Grothendieck Ring of Varieties

Specialeforsvar ved Johan Rydholm

Titel: The Grothendieck Ring of Varieties

Abstract: We give an introduction to the Grothendieck ring of varieties over a field, which is the universal multiplicative invariant. We review some of the questions and important results having been obtained. In particular, in the first part: we give Bittner’s proof of the presentation of the Grothendieck ring by Blow-up relations; we give a proof of Larsen and Lunts result that the Grothendieck ring modulo the affine line is the free Abelian group on stably birational classes of smooth and projective varieties; we sketch some constructions of zero-divisors in the Grothendieck ring, in particular we give Borisov’s proof that the affine line is a zero-divisor, and why this gives a negative answer to paste-and-cut conjecture; we review the positive (partial) answers to the pasteand- cut conjecture in a paper of Liu and Sebag; we introduce Kapranov’s zeta function and review most of the proofs for its rationality for curves, and rationality and non-rationality, respectively, for surfaces, depending on the Kodaira dimension. In the second part we introduce the $G$-equivariant Grothendieck ring of varieties over a field with an action of a finite Abelian group. We sketch the result that the Bittner presentations can be given also in $G$-equivariant case; we introduce the zeta-function for a variety with a $G$-action, and we extend Kapronov’s result that the zeta-functions of curves are rational to hold also when there are $G$-actions on the curves.

Vejleder: Lars Halvard Halle
Censor: Pieter Beelen