Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ be a projective bundle, where $\mathcal{E}$ is a rank two vector bundle over $\mathbb{P}^n$.

If $n = 0$ then $X = \mathbb{P}^1$, and for $n = 1$ we have that $X$ is a Hirzebruch surface. In both cases $-K_X$ is effective.

Is there an example, with $n\geq 2$, of a projective bundle $X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ such that $-K_X$ is not effective?


1 Answer 1


There is a formula, $K_X\simeq \mathcal{O}_X(-2)\otimes \pi^*\det \mathcal{E}^{\vee}\otimes \pi^*K_{\mathbb{P}^n}$.

(More generally, for rank $r+1$ bundle, replace -2 by $-r-1$ in the above formula.)

Now $\pi^*K_{\mathbb{P}^n}\simeq \pi^*\mathcal{O}_{\mathbb{P}^n}(-n-1).$ So $$K_X\simeq \mathcal{O}_X(-2)\otimes \pi^*\det \mathcal{E}^{\vee}\otimes\pi^*\mathcal{O}_{\mathbb{P}^n}(-n-1)=\mathcal{O}_X(-2)\otimes \pi^*(\det \mathcal{E}^{\vee}\otimes\mathcal{O}_{\mathbb{P}^n}(-n-1))$$

Hence $$-K_X\simeq \mathcal{O}_X(2)\otimes \pi^*(\det \mathcal{E}\otimes\mathcal{O}_{\mathbb{P}^n}(n+1))$$

So $-K_X$ is always effective as long as the line bundle on the RHS above has a non-zero section.

Added (I will leave this here for future readers): When I wrote this, I wasn't aware of Hartshorne's conjecture for rank 2 bundles. Thanks to @YosemiteStan for the enlightenment! See YosemiteStan's and abx's comment below for more details.

  • $\begingroup$ Thank you. Twisting $\mathcal{E}$ by a sufficiently positive line bundle on $\mathbb{P}^n$ couldn't we make $\text{det}(\mathcal{E})$ positive? If so it seems that $-K_X$ is always effective independently from what $\mathcal{E}\rightarrow\mathbb{P}^n$ is. Is this the case? $\endgroup$
    – GDR
    Aug 6, 2021 at 17:45
  • $\begingroup$ @GDR Yes. That is correct. Projectivization of $\mathcal{E}$ and that of twist of $\mathcal{E}$ by a line bundle are isomorphic. So $-K_X$ is always effective. $\endgroup$ Aug 6, 2021 at 17:47
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    $\begingroup$ @Evans Gambit: This is definitely wrong. You are confusing the tautological line bundle $\mathscr{O}_X(1)$ and the pull back $\pi ^*\mathscr{O}_{\mathbb{P}^n}(1)$. $\endgroup$
    – abx
    Aug 6, 2021 at 19:04
  • $\begingroup$ @abx Thanks a lot for pointing out the mistake. I have made the correction. Let me know if you spot anything other errors. $\endgroup$ Aug 6, 2021 at 20:41
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    $\begingroup$ If you push forward the right hand side of the final equation you should get (I think??) $\mathrm{Sym}^2(\mathcal{E}^\vee)\otimes \mathrm{det}(\mathcal{E})\otimes \mathcal{O}(n+1)$. When $\mathcal{E}$ is a sum of line bundles it looks like that always has a section (another way to see this should be to observe that these projective bundles are toric I think). On the other hand, I doubt these always have a section when n=2, but if you believe Hartshorne's conjecture on rank 2 bundles then perhaps it is true when n is at least 7. $\endgroup$ Aug 6, 2021 at 21:02

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