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Im not sure if this counts as a full answer, but it is a nice example which will hopefully shed light on some of your questions.

The canonical bundle $\omega$ of an Enriques surface satisfies $\omega \otimes \omega=0$, but $\omega\neq 0$ in the Picard group. It follows that $\omega$ is not the restriction of any line bundle in $\mathbb{P}^n$, as these can't be non-zero torsion.

Im not sure if this counts as a full answer, but it is a nice example which will hopefully shed light on some of your questions.

The canonical bundle $\omega_X$ of an Enriques surface $X$ satisfies $\omega_X \otimes \omega_X=\mathcal{O}_X$, but $\omega_X\neq \mathcal{O}_X$ in the Picard group. It follows that $\omega_X$ is not the restriction of any line bundle in $\mathbb{P}^n$, as these can't be non-zero torsion.

Im not sure if this counts as a full answer, but it is a nice example which will hopefully shed light on some of your questions.

The canonical divisor $K$ of an Enriques surface satisfies $2K=0$, but $K\neq 0$ in the Picard group. It follows that $K$ is not the restriction of any divisor in $\mathbb{P}^n$, as these are necessarily not torsion.

Im not sure if this counts as a full answer, but it is a nice example which will hopefully shed light on some of your questions.

The canonical bundle $\omega$ of an Enriques surface satisfies $\omega \otimes \omega=0$, but $\omega\neq 0$ in the Picard group. It follows that $\omega$ is not the restriction of any line bundle in $\mathbb{P}^n$, as these can't be non-zero torsion.

Im not sure if this counts as a full answer, but it is a nice example which will hopefully shed light on some of your questions.

The canonical divisor $K$ of an Enriques surface satisfies $2K=0$, but $K\neq 0$ in the Picard group. It follows that $K$ is not the restriction of any divisor in $\mathbb{P}^n$, as these are neccessarily not torsion.

Im not sure if this counts as a full answer, but it is a nice example which will hopefully shed light on some of your questions.

The canonical divisor $K$ of an Enriques surface satisfies $2K=0$, but $K\neq 0$ in the Picard group. It follows that $K$ is not the restriction of any divisor in $\mathbb{P}^n$, as these are necessarily not torsion.