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Let $\pi: X\rightarrow \mathbb{P_\mathbb{R}^1}$ be a conic bundle ($\mathbb{R}$-minimal) with 3 reducible fibres. Consider $X$ as $G$-surface where $G\subset Aut_\mathbb{R}(X)$ is a group of order $5$. Hence $G$ acts on $\mathbb{P_\mathbb{R}^1}$.

How can I see that $G$ has at least $3$ fixed points on $\mathbb{P_\mathbb{R}^1}$?

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$G$ takes a reducible fiber to a reducible fiber. Hence the corresponding three points of $P^1$ form a union of $G$-orbits. But any nontrivial $G$-orbit has length $5$, hence each of these three points is fixed.

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  • $\begingroup$ True. And moreover, since $G$ fixes three points on $\mathbb{P}^1$, it acts trivially on it. Moreover, it preserves the two component of any reducible fibre. Seems that the situation is quite not natural at all. $\endgroup$
    – Jérémy Blanc
    Dec 9, 2012 at 15:42

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