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Let $X$ be a Fano manifold of Picard number one. It is known that $X$ is rationally connected. A component $K$ of rational curves in $X$ is called a minimal rational component if the evaluation map $\mathbb{P}^1 \times K \to X$ is dominant and the degree of the curves in $K$ is minimal with this property. A curve in such a component is called a minimal rational curve.

Question: Is it true that any two points $X$ can be connected by a chain of minimal rational curves?

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    $\begingroup$ With your revised formulation, the answer is yes. This is a theorem of Alan Nadel. $\endgroup$ Apr 16, 2022 at 11:54
  • $\begingroup$ I attributed the result to Nadel, but I believe it is independently due to Frederic Campana. This is how Nadel and Campana proved boundedness of deformation types of complex Fano manifolds of Picard number one (and fixed dimension). $\endgroup$ Apr 16, 2022 at 16:01

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No. For instance, if $X$ is a Fano threefold of index 1 (e.g. a quartic hypersurface in $\mathbb{P}^4$), minimal curves are lines, and they only sweep a divisor in $X$. So, two points outside of this divisor cannot be connected by a chain of lines.

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  • $\begingroup$ Here, by minimal rational curve, I mean curves in a dominating minimal rational component. We say a component $K$ of rational curves is dominating if the evaluation map $\mathbb{P}^1 \times K \to X$ is dominant. We say it is a minimal rational component if the degree of the curves is minimal with this property. In particular, $X$ can be covered by minimal rational curves. $\endgroup$
    – LAPRAS
    Apr 16, 2022 at 6:57
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    $\begingroup$ I wouldn't call this "minimal", but maybe "minimal dominating" is OK. And anyway, it is better to clarify your question in this aspect. $\endgroup$
    – Sasha
    Apr 16, 2022 at 7:01
  • $\begingroup$ Ok. I will do that. What would be the answer for this case? $\endgroup$
    – LAPRAS
    Apr 16, 2022 at 7:03

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