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Many concepts in geometry apply a priori specifically to line bundles. For certain theories like ampleness, nef, positivity I know that in order to generalize to arbitrary ranked vector bundles $E\to X$ we use the tautological line bundle $\mathcal{O}_E(-1)\to \mathbb{P}(E)$ over the projectivization of $E$. My questions are the following

  • Are there other theories for which we also use $\mathcal{O}_E(-1)\to \mathbb{P}(E)$?
  • "Why" does this technique work? In other words what essential characteristics of $E$ are captured in $\mathcal{O}_E(-1)$?
  • Are there other ways to reduce a vector bundle to a line bundle and if yes for which concepts?
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    $\begingroup$ Atiyah thought about vector bundles on elliptic curves by finding subbundles and quotient bundles until you get down to line bundles. $\endgroup$
    – Ben McKay
    Commented Jan 11, 2022 at 10:33
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    $\begingroup$ Your third bullet point is the splitting principle. $\endgroup$
    – Branimir Ćaćić
    Commented Jan 11, 2022 at 11:04
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    $\begingroup$ @BranimirĆaćić, the third bullet point is "Are there other ways …?" Did you mean the second bullet point? $\endgroup$
    – LSpice
    Commented Jan 11, 2022 at 13:07
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    $\begingroup$ Argh, you’re absolutely right. #2 is the splitting principle, #3 is whether or not there are other techniques like the splitting principle. $\endgroup$
    – Branimir Ćaćić
    Commented Jan 11, 2022 at 13:44
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    $\begingroup$ I don't mean to be too pick but for positivity etc. you need $\mathcal{O}_E(+1)$ and depending on your convention, you need to work with $\mathbb{P}(E)$ or $\mathbb{P}(E^\vee)$. The key point is that you want $\pi_*\mathcal{O}_E(1)=E$. This partly answers why it works. $\endgroup$
    – Donu Arapura
    Commented Jan 11, 2022 at 14:02

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