ScalingStacks

Theorem 9.3 . [030B]

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Theorem 9.3.

Given a stable log map (C†/W†,f)(C^{\dagger}/W^{\dagger},f), there is a basic stable log map (Cb†/Wb†,fb)(C_{b}^{\dagger}/W_{b}^{\dagger},f_{b}) fitting into a commutative diagram

C†\textstyle{C^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Cb†\textstyle{C_{b}^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}X†\textstyle{X^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}W†\textstyle{W^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Wb†\textstyle{W_{b}^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}S†\textstyle{S^{\dagger}}

where the left-hand square is cartesian in the category of fine saturated log schemes and the maps W→WbW\rightarrow W_{b} and C→CbC\rightarrow C_{b} of underlying schemes are isomorphisms. Furthermore, (Cb†/Wb†,fb)(C_{b}^{\dagger}/W_{b}^{\dagger},f_{b}) and the maps in the above diagram are determined by (C†/W†,f)(C^{\dagger}/W^{\dagger},f) uniquely up to unique isomorphism.

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