Seminar Algebra and Geometry: Marcello Bernardara (Toulouse)

From noncommutative motivic measures to subgroups of the Cremona group

Let X be a smooth projective variety over a field k, and assume that weak factorization holds (e.g., k has characteristic zero).
I will introduce the Grothendieck ring of triangulated categories, and show how, using Bondal-Larsen-Lunts motivic measure, a subgroup of such ring will define a subgroup of the group Bir(X) of birational self-maps of X. A main example is given by the filtration via the motivic dimension, which induces a filtration on Bir(X). As a consequence, in the case X=Pⁿ, we can show that the group generated by the standard Cremona transformation and PGL(n+1) is strictly contained in the group contracting rational varieties, as soon as n > 4. Another example allows to reconstruct Frumkin's genus filtration of Bir(X) in the cases where X is a uniruled threefold.

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