The invertible-sheaves tag has no usage guidance.

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### What can be said about a pullback of a very ample line bundle w.r.t birational maps?

Let $X$ be a smooth projective variety and $\phi: X \to \mathbb P^n$ be a map. If $\phi$ is an embedding then $E=\phi^*(O(1))$ is very ample. But can one say something if $\phi$
is birational (but not ...

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### Justification of the term “invertible sheaf”

Let $X$ be a locally ringed space (or a scheme) and $M,N$ two $\mathcal{O}_X$-modules such that $M \otimes N \cong \mathcal{O}_X$. Does it follow that $M$ is invertible in the usual sense, namely that ...