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Let $X$ be a smooth projective variety of dimension $n \geq 2$. $H$ be the very ample divisor on $X$ giving the embedding. Let $S$ be a general surface section under $H$. Then we know that $K_S:=k_X|_S +c_1(\mathcal N_{S/X})$, where $\mathcal N_{S/X}$ stands for the normal bundle. Then I have the following questions:

$(1)$ Is it true that $c_1(\mathcal N_{S/X}) = (n-2)H|_S$?

$(2)$ If $K_X+(n-2)H$ is nef and $(1)$ is true, then can we say that $K_S$ is nef because pullback of a nef divisor under a finite morphism (here inclusion) is still nef? Moreover is $K_S$ automatically an effective divsior (allowing $0$)?

$(3)$ If $K_S$ is nef, then can we say that $S$ can't be rational? ( I can see this for specific rational surfaces e.g. $\mathbb P^2$, Hirzebruch surfaces that they dont have a nef camonical divisor.. but is there a general way to see this?)

Any insight is welcome

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    $\begingroup$ (1) Yes. The normal bundle is $H^{{\scriptscriptstyle\oplus }(n-2)}.\hskip3.5cm$ (2) Yes, $K_S$ is nef; no, it is not necessarily effective.$\hskip1.3cm$ (3) $K_S$ is nef if and only if $S$ is minimal of Kodaira dimension $\geq 0.\hskip9cm$ This being said, your questions would be more appropriate at MSE. $\endgroup$
    – abx
    Commented Nov 14, 2023 at 14:18
  • $\begingroup$ @abx, Could you please briefly indicate if $(1)$ follows directly by definition or some exact sequence involving normal bundle, if the argument given for $(2)$ is correct? $\endgroup$
    – Proj
    Commented Nov 15, 2023 at 14:20
  • $\begingroup$ (1) follows from $\mathscr{N}_{S/X}^*\cong \mathscr{I}/\mathscr{I}^2$, where $\mathscr{I}$ is the ideal sheaf of $S$ in $X$. Yes, your argument for (2) is correct. $\endgroup$
    – abx
    Commented Nov 15, 2023 at 18:01

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