I was (trying to) read through Shinichi Mochizuki's paper The Geometry Of the Compactification of the Hurwitz Scheme and was stumped within just a few paragraphs by the function he defines $f : C \rightarrow \textbf{P}^1$. What is $C$ and what is $\textbf{P}^1$?
Also, what does $\textbf{Z}[\frac{1}{b!}]$ mean (referenced in the first theorem of the introduction)?