# Minimally rationally connectedness of Fano manifold

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?

• With your revised formulation, the answer is yes. This is a theorem of Alan Nadel. Apr 16, 2022 at 11:54
• 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). Apr 16, 2022 at 16:01

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.

• 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. Apr 16, 2022 at 6:57
• I wouldn't call this "minimal", but maybe "minimal dominating" is OK. And anyway, it is better to clarify your question in this aspect. Apr 16, 2022 at 7:01
• Ok. I will do that. What would be the answer for this case? Apr 16, 2022 at 7:03