KUMMER COVERINGS AND SPECIALISATION

We prove versions of various classical results on specialisation of fundamental groups in the context of log schemes in the sense of Fontaine and Illusie, generalising earlier results of Hoshi, Lepage and Orgogozo. The key technical result relates the category of finite Kummer étale covers of an fs log scheme over a complete Noetherian local ring to the Kummer étale coverings of its reduction.

Logarithmically regular morphisms

We consider the stack $${\mathcal {L}}og_{X}$$ L o g X parametrizing log schemes over a log scheme X, and weak and strong properties of log morphisms via $${\mathcal {L}}og_{X}$$ L o g X , as defined by Olsson. We give a concrete combinatorial presentation of $${\mathcal {L}}og_{X}$$ L o g X , and prove a simple criterion of when weak and strong properties of log morphisms coincide. We then apply this result to the study of logarithmic regularity, derive its main properties, and give a chart criterion analogous to Kato’s chart criterion of logarithmic smoothness.

Notes on algebraic log stack

Let [Formula: see text] be a stack over the category of fine log schemes. If [Formula: see text] has a representable fppf covering, then, it has enough compatible minimal objects. As a consequence, we prove the equivalence between two notions of log moduli stacks which appear in literatures. Also, we obtain several fundamental results of algebraic log stacks.