# Maximal closed subscheme over which a line bundle is trivial

The following statement comes from the book "Abelian varieties" by Mumford, at the very beginning of chapter 10. All varieties/schemes are defined over a field $$k$$.

Let $$X$$ be a complete variety, $$Y$$ any scheme and $$\mathcal L$$ a line bundle on $$X\times Y$$. Then there exists a unique closed subschemes $$Y_1 \hookrightarrow Y$$ such that the restriction of $$\mathcal L$$ to $$X\times Y_1$$ is isomorphic to the pullback of a line bundle $$\mathcal M$$ on $$Y_1$$ (via the projection morphism) ; and such that $$Y_1$$ is maximal with respect to this property.

This closed subscheme $$Y_1$$ is called the maximal closed subscheme of $$Y$$ over which $$\mathcal L$$ is trivial.

Now, this may be a silly question, but to my understanding we usually call a line bundle trivial when it is isomorphic to the structure sheaf of the scheme. With this in mind, why wouldn't we require the condition "the restriction of $$\mathcal L$$ to $$X\times Y_1$$ is isomorphic to $$\mathcal O_{X\times Y_1}$$" instead ? It doesn't seem equivalent as the line bundle $$\mathcal M$$ may not be trivial. Is there a specific reason for this ?

$$\newcommand{\L}{\mathcal L} \newcommand{\M}{\mathcal M}$$I believe that in this context you are supposed to think of $$\L$$ as a family of line bundles on $$X$$ parametrized by $$Y$$. For every $$k$$-point $$i:y\to Y$$ you can take the pullback $$\widetilde i:X\to X\times Y$$, and obtain a line bundle $$\widetilde i^*\L$$ on $$X$$. The proposition is stating that $$Y_1$$ is such that whenever $$y\in Y_1$$, then $$\widetilde i^*\L\cong \mathcal O_X$$.
If $$Y$$ is reduced, then being trivial on every fiber is equivalent to being a pullback from a line bundle on $$Y$$. Let us draw this diagram: $$\require{AMScd} \begin{CD} X @>{\widetilde i}>> X\times Y\\ @V{p}VV @V{\pi}VV \\ y @>{i}>> Y. \end{CD}$$ First, if $$\L = \pi^*\M$$ for some line bundle $$\M$$, then $$\widetilde i^*\L = p^*i^*\M$$, which is trivial, since it's pulled back from a point (this implication holds for any $$Y$$).
To prove the converse we can use this proposition you are asking about in Mumford's book. Suppose $$\widetilde i^* Y$$ is trivial for every $$y$$. By the proposition, there is a maximal closed subscheme $$Y_1$$ such that $$\L|_{X\times Y_1}$$ is a pullback of some line bundle $$\M$$ on $$X\times Y_1$$. Every point of $$Y$$ is a (non-maximal) closed subscheme with this property, so $$Y_1$$ must contain all the points of $$Y$$, and since $$Y$$ is reduced, $$Y_1=Y$$.
If you interpret the statement as $$\L$$ being trivial on $$X\times Y_1$$, then it's false. In the simplest example, take $$X$$ to be a point, and let $$Y$$ be such that it has a nontrivial line bundle $$\L$$ (which we can think of as a line bundle on $$X\times Y$$), for example $$Y=\mathbb P^1$$. Then over every point of $$Y$$, $$\L$$ is trivial, yet globally it is not, so there cannot exist such a biggest closed subscheme.