Logarithmic Picard groups,chip firing,and the combinatorial rank

Mathematische Zeitschrift - Illusie has suggested that one should think of the classifying group of $$M_X^{gp}$$ -torsors on a logarithmically smooth curve X over a standard logarithmic point as a...     

Clutching and gluing in tropical and logarithmic geometry

The classical clutching and gluing maps between the moduli stacks of stable marked algebraic curves are not logarithmic, i.e. they do not induce morphisms over the category of logarithmic schemes, since they factor through the boundary. Using insight from tropical geometry, we enrich the category of logarithmic schemes to include so-called sub-logarithmic morphisms and show that the clutching and gluing maps are naturally sub-logarithmic. Building on the recent framework developed by Cavalieri, Chan, Wise, and the third author, we further develop a stack-theoretic counterpart of these maps in the tropical world and show that the resulting maps naturally commute with the process of tropicalization.