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In Starr's paper https://www.math.stonybrook.edu/~jstarr/papers/moduli4.pdf the folk result that the fibred category of pairs $(X\to S, L)$, where $S$ is an affine scheme, $X\to S$ is flat proper finitely presented morphism of algebraic spaces and $L$ is a relatively ample line bundle on $X$, is an algebraic stack is proven.

This made me wonder:

Is the fibred category of pairs $(X\to S,L)$, where $X\to S$ is flat proper finitely presented and $L$ is a relatively big line bundle on $X$, an algebraic stack?

I couldn't find an answer in the stacks project.

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    $\begingroup$ I just want to say, all of the results in the first sections of that article are now subsumed by the many (fantastic) articles of Hall and Rydh. Also all of the results and counterexamples from the last part of that article are folk results. Having said all of that, I am skeptical that existence of a big line bundle is sufficient for algebraicity, but I will have to think about it. $\endgroup$ Commented Jul 30, 2015 at 17:45
  • $\begingroup$ @JasonStarr I slightly rewrote the question to make it clear that the result I mention is folklore. I tried finding a variety of general type with no effective formal versal deformation but haven't succeeded so far... $\endgroup$
    – Pancho
    Commented Jul 30, 2015 at 18:58

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