Motivic invariants of birational maps

TL;DR

This paper introduces new invariants for birational maps using algebraic structures like the Kontsevich--Tschinkel group, providing insights into the Grothendieck ring and Cremona groups.

Contribution

It constructs novel invariants of birational maps valued in advanced algebraic groups, enhancing understanding of the Grothendieck ring and L-equivalence.

Findings

New invariants for birational maps are constructed.

Unexpected results about Cremona groups are proved.

Invariants help analyze the structure of the Grothendieck ring.

Abstract

We construct invariants of birational maps with values in the Kontsevich--Tschinkel group and in the truncated Grothendieck groups of varieties. These invariants are morphisms of groupoids and are well-suited to investigating the structure of the Grothendieck ring and L-equivalence. Building on known constructions of L-equivalence, we prove new unexpected results about Cremona groups.