I am trying to understand Mumford's statement of the Seesaw theorem given at the start of Chapter 3 in his book Abelian Varieties.
The statement is: Let $X$ be a complete variety, $Y$ a scheme (both over an algebraically closed field $k$) and $L$ a line bundle on $X \times Y$. Then there exists a unique closed subscheme $Z$ of $Y$ such that $L|_{X \times Z} = p_{Z}^{*}N$ for some line bundle $N$ on $Z$ and if $f \colon S \rightarrow Y$ such that $(1_{X},f)^{*}L = p_{S}^{*}M$ then $f$ factors through $Z$.
My questions are: What is $Z$ if $L = p_{X}^{*}N$ for $N$ a non-trivial line bundle on $X$ - is it $Spec(k)$? If so how does one prove it?
Intuitively, it seems to me that it should be $Spec(k)$, because the data of the line bundle all comes from $X$ so there should be no non-trivial subscheme of $Y$ over which $L$ is pulled back from.