# Divisors on projective bundles

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?

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.

• 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?
– GDR
Aug 6, 2021 at 17:45
• @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. Aug 6, 2021 at 17:47
• @Evans Gambit: This is definitely wrong. You are confusing the tautological line bundle $\mathscr{O}_X(1)$ and the pull back $\pi ^*\mathscr{O}_{\mathbb{P}^n}(1)$.
– abx
Aug 6, 2021 at 19:04
• @abx Thanks a lot for pointing out the mistake. I have made the correction. Let me know if you spot anything other errors. Aug 6, 2021 at 20:41
• 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. Aug 6, 2021 at 21:02