Let $X$ be a smooth projective variety over a field $k$. Let $L$ be an invertible sheaf on $X$. Suppose that $L$ is big and globally generated. Can one conclude that the associated morphism $\phi_L : X \to \mathbb{P}^n$ is generically finite? Thanks in advance.
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$\begingroup$ Welcome new contributor. Yes, the morphism $\phi_L$ is generically finite to its image. $\endgroup$– Jason StarrCommented Jun 5, 2022 at 1:25
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$\begingroup$ @JasonStarr Thanks for the prompt reply! Do you happen to know a reference? $\endgroup$– MaquiCommented Jun 5, 2022 at 1:28
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5$\begingroup$ I don't see it in Positivity, but one way to see it is this, let $Z = \phi_L(X)$, let $\eta$ be the generic point of $Z$ and let $f : X_{\eta} \to \eta$ be the base change. Note $L|_{X_{\eta}}$ is still big and globally generated. Furthermore $X_{\eta}$ is a projective variety over the field $k(\eta)$ and $L|_{X_{\eta}} = \phi_L^* O(1)|_{X_{\eta}} = f^* k(\eta) = O_{X_{\eta}}$. Now the only time the structure sheaf of a projective variety can be big is if the variety is 0-dimensional, so $X_{\eta}$ is zero dimensional, thus $X \to Z$ is generically finite. $\endgroup$– Karl SchwedeCommented Jun 5, 2022 at 3:38
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$\begingroup$ @KarlSchwede Thanks! if you post this as an answer I'd be happy to accept it. $\endgroup$– MaquiCommented Jun 5, 2022 at 20:37
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2$\begingroup$ You can also see it because if $\phi_L$ maps to something lower-dimensional, then $0=f^*(O(1))^n$, so $L^n=0$. However, since $L$ is globally generated (hence nef), it is big if and only if $L^n>0$. $\endgroup$– Ennio Mori coneCommented Jun 7, 2022 at 10:02
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I don't see it in Lazarsfeld's Positivity, but one way to see it is this, let $Z=\phi_L(X)$, let $\eta$ be the generic point of $Z$ and let $f:X_{\eta}→\eta$ be the base change. Note $L|_{X_{\eta}}$ is still big and globally generated. Furthermore $X_\eta$ is a projective variety over the field $k(\eta)$ and $$L|_{X_{\eta}}=\phi^* \mathcal{O}(1)|_{X_{\eta}}=f^*k(η)=\mathcal{O}_{X_η}.$$ Now the only time the structure sheaf of a projective variety can be big is if the variety is 0-dimensional, so $X_{\eta}$ is zero dimensional, thus $X \to Z$ is generically finite.
answered Jun 6, 2022 at 19:26