Consider a 2-dimensional smooth projective algebraic surface S over complex numbers. Could you recommend any exact references to the proofs of the following assertions (of course, if they are true):

If S is rationally chain connected then S is rational.

If through each point of S one can draw a rational curve then S is uniruled.

If S is uniruled then S is ruled.

Recall that a surface is *rationally chain connected*, through each generic pair of points can be joined by a chain of rational curves on the surface. A *uniruled* surface is the one which admits a dominant map from $X\times \mathbb{P}^1$ for some curve $X$. A *ruled* surface is the one which admits a birational map from $X\times \mathbb{P}^1$ for some curve $X$.

Assertion 1 = [1,Proposition IV.3.6] + [1,Theorem IV.3.10] + [1,Proposition IV.3.3.1] + [1,Excercise IV.3.3.5], the latter given without proof. Cf. [1, Excercise IV.3.12.2]. Although this approach seems to be an overkill.

[1] J. Kollar, Rational curves on algebraic varieties, Springer-Verlag, Berlin–Heidelberg, 1996.

**Edit**

[2] A.Beauville, Complex algebraic surfaces,London Math. Soc. Student Texts 34, 2nd edition, Cambridge Univ. Press, 1996, 132p.

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