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This survey paper reports on work of Birkar, who confirmed a long-standingconjecture of Alexeev and Borisov-Borisov: Fano varieties with mildsingularities form a bounded family once their dimension is fixed. FollowingProkhorov-Shramov, we explain how this boundedness result implies thatbirational automorphism groups of projective spaces satisfy the Jordanproperty, answering a question of Serre in the positive.

Boundedness results for singular Fano varieties and applications to Cremona groups