3
$\begingroup$

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 $E$.

$\endgroup$
4
  • 4
    $\begingroup$ 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
    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
    Jan 13, 2011 at 14:58
  • $\begingroup$ @Artie: Thanks for your example, I do want $X$ to have dimension more than 2. $\endgroup$ Jan 13, 2011 at 17:21
  • $\begingroup$ @Artie: You are right about the "more specifically". I have edited the question accordingly. $\endgroup$ Jan 13, 2011 at 17:27

1 Answer 1

5
$\begingroup$

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 $\pi$.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.