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.
J. Math. Soc. Japan
73(3):
861-883
(July, 2021).
DOI: 10.2969/jmsj/82658265
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