Using a filtration on the Grothendieck ring of triangulated categories, we define the motivic categorical dimension of a birational map between smooth projective varieties. We show that birational transformations of bounded motivic categorical dimension form subgroups, which provide a nontrivial filtration of the Cremona group. We discuss some geometrical aspect and some explicit example. We can moreover define, in some cases, the genus of a birational transformation, and compare it to the one defined by Frumkin in the case of threefolds.
" Categorical dimension of birational transformations and filtrations of Cremona groups ." J. Math. Soc. Japan Advance Publication 1 - 23, April, 2021. https://doi.org/10.2969/jmsj/82658265