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.



Morphism attached to a big and globally generated line bundle
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.


