Toric singularities revisited

In [K. Kato, Toric singularities, Amer. J. Math. 116 (5) (1994) 1073–1099], Kato defined his notion of a log regular scheme and studied the local behavior of such schemes. A toric variety equipped with its canonical logarithmic structure is log regular. And, these schemes allow one to generalize toric geometry to a theory that does not require a base field. This paper will extend this theory by removing normality requirements.