# On line bundles in derived algebraic geometry

## TL;DR

This paper provides examples in derived algebraic geometry where line bundles on the truncation do not extend to the original derived scheme, highlighting limitations of the pullback map between Picard groups.

## Contribution

It constructs explicit examples of derived schemes with non-surjective Picard pullback maps, showing that truncations can have line bundles not extendable to the original scheme.

## Key findings

- Examples of derived schemes with non-surjective Picard pullback maps
- Truncations are projective hypersurfaces with trivial line bundles
- Original schemes lack nontrivial line bundles and are not quasi-projective

## Abstract

We give examples of derived schemes $X$ and a line bundle $\Ls$ on the truncation $tX$ so that $\Ls$ does not extend to the original derived scheme $X$. In other words the pullback map $\Pic(X) \to \Pic(tX)$ is not surjective. Our examples have the further property that, while their truncations are projective hypersurfaces, they fail to have any nontrivial line bundles, and hence they are not quasi-projective.