Is there a lower bound on the slope (i.e. ratio of degree to rank) of normal bundles of smooth projective curves embedded in smooth projective varieties?
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$\begingroup$ It looks like Sasha has answered the question. I just wanted to point out that if you instead ask whether on a fixed variety $X$, there exists a bound that holds for all curves on $X$, I think this is a hard question; it seems like a reasonable analog of the bounded negativity conjecture to higher dimensions. $\endgroup$– John LCommented Sep 4, 2021 at 1:49
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No. For instance the Hirzebruch surface $F_n$ contains a smooth rational curve with normal bundle $\mathcal{O}(-n)$ of slope $-n$.
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$\begingroup$ What if we ask the ambient variety to have non-negative Kodaira dimension? $\endgroup$– WenlongCommented Sep 3, 2021 at 13:28
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3$\begingroup$ @Wenlong Welcome new contributor. That is still negative. Fix a line in projective $3$-space. For every positive integer $d$ you can find a smooth degree-$d$ hypersurface containing the line. The normal bundle is an invertible sheaf on the line of degree $2-d$, which is arbitrarily negative as $d$ increases. $\endgroup$ Commented Sep 3, 2021 at 13:42
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$\begingroup$ @JasonStarr Thank you. I hope it doesn't annoy you but what if we restrict to Calabi-Yau threefolds? $\endgroup$– WenlongCommented Sep 3, 2021 at 13:52