Logarithmic geometry in the sense of Fontaine-Kato-Illusie seeks to extend the beautiful combinatorial techniques available for toric varieties to all schemes by providing local monoidal coordinates on their structure sheaves. Using logarithmic geometry we may, in particular, categorify the well-known notion of toroidal embeddings, which has proved incredibly useful to the study of degenerations of algebraic varieties (e.g. when proving semistable reduction etc.).

The purpose of this talk is to give an introduction to logarithmic geometry, only assuming basic toric geometry on the level of the first chapter of Fulton's book. Armed with these techniques, we can then explore one of the most immediate applications of these techniques, the construction of the moduli space of logarithmically smooth curves as well as its connection to the classical Deligne-Knudsen-Mumford moduli space of stable curves with marked points. Speaker(s): Martin Ulirsch (UM)