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LSpice
LSpice
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From$\DeclareMathOperator\Log{Log}$From https://arxiv.org/pdf/1103.2140.pdfGilliam - Logarithmic stacks and minimality:

A category fibered in groupoids (CFG) over schemes with a map to Olsson's stack $Log$$\Log$ induces a CFG over log schemes. This is fully faithful, but there are loads of other CFG's over log schemes that do not come from this process.

To determine which ones do, note that if a CFG has a map $X \to Log$$X \to \Log$, any scheme mapping to $X$ may be given the pulled-back "minimal" log structure. Among the log schemes mapping to $X$ with the same underlying scheme, there's a final one whose map to $X$ is strict. There's a categorical way to describe and detect minimality.

The map $X \to Log$$X \to \Log$ could parametrize a Cartier divisor as you suggest, or more generally any log structure. Log structures are essentially given by systems of line bundles with sections with some relations among their tensor powers (you need "quasi-integrality" here).

From https://arxiv.org/pdf/1103.2140.pdf:

A category fibered in groupoids (CFG) over schemes with a map to Olsson's stack $Log$ induces a CFG over log schemes. This is fully faithful, but there are loads of other CFG's over log schemes that do not come from this process.

To determine which ones do, note that if a CFG has a map $X \to Log$, any scheme mapping to $X$ may be given the pulled-back "minimal" log structure. Among the log schemes mapping to $X$ with the same underlying scheme, there's a final one whose map to $X$ is strict. There's a categorical way to describe and detect minimality.

The map $X \to Log$ could parametrize a Cartier divisor as you suggest, or more generally any log structure. Log structures are essentially given by systems of line bundles with sections with some relations among their tensor powers (you need "quasi-integrality" here).

$\DeclareMathOperator\Log{Log}$From Gilliam - Logarithmic stacks and minimality:

A category fibered in groupoids (CFG) over schemes with a map to Olsson's stack $\Log$ induces a CFG over log schemes. This is fully faithful, but there are loads of other CFG's over log schemes that do not come from this process.

To determine which ones do, note that if a CFG has a map $X \to \Log$, any scheme mapping to $X$ may be given the pulled-back "minimal" log structure. Among the log schemes mapping to $X$ with the same underlying scheme, there's a final one whose map to $X$ is strict. There's a categorical way to describe and detect minimality.

The map $X \to \Log$ could parametrize a Cartier divisor as you suggest, or more generally any log structure. Log structures are essentially given by systems of line bundles with sections with some relations among their tensor powers (you need "quasi-integrality" here).

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Leo Herr
Leo Herr
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From https://arxiv.org/pdf/1103.2140.pdf:

A category fibered in groupoids (CFG) over schemes with a map to Olsson's stack $Log$ induces a CFG over log schemes. This is fully faithful, but there are loads of other CFG's over log schemes that do not come from this process.

To determine which ones do, note that if a CFG has a map $X \to Log$, any scheme mapping to $X$ may be given the pulled-back "minimal" log structure. Among the log schemes mapping to $X$ with the same underlying scheme, there's a final one whose map to $X$ is strict. There's a categorical way to describe and detect minimality.

The map $X \to Log$ could parametrize a Cartier divisor as you suggest, or more generally any log structure. Log structures are essentially given by systems of line bundles with sections with some relations among their tensor powers (you need "quasi-integrality" here).