Simple relations in the Cremona group

We give a simple set of generators and relations for the Cremona group of the plane. Namely, we show that the Cremona group is the amalgamated product of the de Jonquières group with the group of automorphisms of the plane, divided by one relation which is $στ=τσ$, where $τ=(x:y:z)\mapsto (y:x:z)$ and $σ=(x:y:z)\dasharrow (yz:xz:xy)$.

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