On birational automorphisms of rational surfaces

Generators and relations of the groups of birational automorphisms are described for six classes of ($k$-minimal) rational surfaces. For surfaces $F$ with $\rho\colon=rk\operatorname{Pic}F=2$, $d\colon=K_F^2=3,4$ and $\rho=1$, $F(k)=\oslash$, $d=9$ relations between the already known generators are described, and for surfaces with $\rho=1$, $F(k)=\oslash$, $d=8,6$ and $\rho=2$, $F(k)=\oslash$, $d=8$ generators and relations are described.