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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 $E$.

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    $\begingroup$ If X is the blowup of P^2 in 9 points, then (-K_X)^2=0. Also, X is rational, hence rationally connected. So the answer to the first question is no. (Maybe you want the dimension of X to be more than 2.) $\endgroup$
    – user5117
    Commented Jan 13, 2011 at 14:55
  • $\begingroup$ Also, the questions in the second paragraph seem to be asking whether every rationally 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..." $\endgroup$
    – user5117
    Commented Jan 13, 2011 at 14:58
  • $\begingroup$ @Artie: Thanks for your example, I do want $X$ to have dimension more than 2. $\endgroup$ Commented Jan 13, 2011 at 17:21
  • $\begingroup$ @Artie: You are right about the "more specifically". I have edited the question accordingly. $\endgroup$ Commented Jan 13, 2011 at 17:27

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It seems to me that the answer to your question is no, because of the following example (which came to my mind after reading Artie Prendergast-Smith's comment).

Consider a pencil $\lambda Q_1 + \mu Q_2$ of quartic surfaces in $\mathbb{P}^3$, and let $Z$ be its base locus, that in general will be a smooth curve of degree $16$. Blowing-up $Z$, we obtain a smooth rationally connected $3$-fold $X$ together with a map $\pi \colon X \to \mathbb{P}^1$, which gives to $X$ the structure of a fibration in $K3$ surfaces. If $F$ is the class of a fibre of $\pi$, the formula for the canonical class of a blow-up yields

$K_X=-F$.

So for every surface $S \subset X$ one has $(-K_X)^2 \cdot S=0$.

This can be obviously generalized in any dimension, by considering a pencil of hypersurfaces of degree $n+1$ in $\mathbb{P}^n$ and blowing-up the corresponding base locus. In this way one obtain a smooth rationally connected $n$-fold $X$ with a fibration $\pi \colon X \to \mathbb{P}^1$ in Calabi-Yau varieties, and the anticanonical divisor of $X$ coincides with a fibre of $\pi$.

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