I've been working through Hartshorne section II.5, and am currently thinking about very ample line bundles.
It's emphasized that very-ampleness of a line bundle is a relative notion, defined in the context of a morphism of schemes $X\to Y$. It seems to me that, in the category of schemes over $Y$, a particularly basic object to consider is the terminal one, the identity morphism $Y\to Y$. So I was led to wonder: what does it mean for a line bundle on $Y$ to be very ample in this context, and does the condition trivialize somehow?
This seems to involve describing sections of the projection $\mathbb{P}^r_Y\to Y$, but I'm not sure how to do this; I'm having difficulty thinking about projective space over any scheme besides Spec of a field, which isn't a particularly interesting case to consider here.