Rational surfaces with a pencil of rational curves and with positive square of the canonical class

In the paper standard $G$-surfaces with a pencil of rational curves and with $(\omega_F\cdot\nobreak\omega_F)>\nobreak0$ are examined up to birational equivalence. It is proved that for $(\omega_F\cdot\omega_F)>1,2,3$ a birational class of these surfaces is uniquely determined by the birational class of their standard pencil of rational curves. For $(\omega_F\cdot\omega_F)>4$ each of these surfaces is birationally equivalent to either the plane $\mathbf P^2$ or some $G$-surface which is a biregular form of the surface $\mathbf P^1\times\mathbf P^1$.
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