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):


1. If S is rationally chain connected then S is rational.
2. If through each point of S one can draw a rational curve then S is uniruled.
3. 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. 



Is each rationally chain connected surface rational?
By the comments of abx you only need an answer to question $3$. You get easily an answer if you assume the results in the classical book of Beauville on algebraic surfaces.


Namely, if $S$ is uniruled, then you can solve the indeterminacy (Theorem II.7) and get a dominant morphism $X \rightarrow S$ where $X$ is birational to a product $C \times {\mathbb P}^1$. It follows, by Proposition III.20, that (being birational to a surface with vanishing plurigenera) all plurigenera of $X$ vanish. The argument of the proof of the same proposition applied to the dominant map $X \rightarrow S$ show that $P\_n(S) \leq P\_n(X)$ and therefore all plurigenera of $S$ vanish too.


Then, if $q=0$, the result follows by Castelnuovo rationality criterion. For the irregular ($q>0$) case, Beuville shows, (Proposition VI.15.(1)) that, if the surface is not ruled, then either $P\_4 \neq 0$ or $P\_6 \neq 0$.


Of course all the mentioned results are highly non trivial, but all proofs are in Beauville's book.


