Does anyone know an example of a rational ruled surface $X=\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(e))$ for $e\ge 0$ which admits a transitive algebraic group action? except the trivial case $\mathbb{P}^1\times\mathbb{P}^1$.
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A rational ruled surface with $e>0$ has a unique irreducible curve with negative selfintersection, so any automorphism has to fix that. Therefore it cannot have a transitive automorphism group. (Actually it also has to fix the ruling, because it has to fix the cone of curves and the negative curve and the fiber of the ruling are the generators, but of course that in itself would allow for a transitive action as the fibers cover the entire surface). 

