Let $X$ be a rationally connected smooth projective variety defined over $\mathbb C$.

(1) Can we find a surface $S \subset X$ such that $ (-K_X)^2 \cdot S > 0 ? $
If yes, can we assume that $S$ intersects properly a given codimension $2$ subvariety $V \subset X$ only at isolated points ? The answer is obviously **no**, as Artie pointed
out in the comments.

(2) Can the square of the first Chern Class of $K_X$ be numerically equivalent to $\sum \lambda_i Y_i$ where $\lambda_i \in \mathbb Q_{<0}$ are negative rational numbers, and $Y_i$ are irreducible codimension two cycles ?

**Edit** : As Artie and Francesco noted, (1) is too much to ask for. I still would like
to know if (2) can hold ?

**Edit 2** : The answer to (2) is yes. If we blow up a point in Francesco's example then
we obtain a $3$-fold $Y$ with $K_Y = -F + 2E$. Thus $K_Y^2$ is numerically equivalent to
$-4 \ell$, where $\ell$ is a line inside the exceptional divisor $E$.

everyrationally connected variety has a certain property, whereas the question in the third paragraph seems to ask if we can find a r.c. variety with a certain property. So I don't quite understand the connective "More specifically..." – Artie Prendergast-Smith Jan 13 '11 at 14:58