Categorical dimension of birational automorphisms and filtrations of Cremona groups

TL;DR

This paper introduces a new categorical dimension for birational maps between smooth projective varieties, leading to a filtration of the Cremona group and connecting to geometric properties like genus.

Contribution

It defines the categorical dimension of birational maps via a filtration on the Grothendieck ring, revealing subgroup structures within the Cremona group and relating to existing geometric invariants.

Findings

Birational automorphisms of bounded categorical dimension form subgroups.

The filtration provides a new perspective on the structure of the Cremona group.

In threefolds, the categorical dimension recovers the genus and relates to Frumkin's filtration.

Abstract

Using a filtration on the Grothendieck ring of triangulated categories, we define the categorical dimension of a birational map between smooth projective varieties. We show that birational automorphisms of bounded categorical dimension form subgroups, which provide a nontrivial filtration of the Cremona group. We discuss some geometrical aspect and some explicit example. In the case of threefolds, we can moreover recover the genus of a birational automorphism, and the filtration defined by Frumkin.