On toric face rings. II (Q1627498)

This paper concerns toric varieties with singularities that are possibly not normal, and even reducible. A semi-log canonical singularity $X$ is defined as a singularity such that a) $X$ is $S_2$ and nodal in codimension one, b) certain pluricanonical sheaves $\omega^{[r]}_{X}$ are invertible, and c) the induced log structure on the normalization has log canonical singularities. The main point is the definition of weakly log canonical singularities by replacing axiom a) with a'): $X$ is $S_2$ and weakly normal. The known pluricanonical sheaves $\omega^{[r]}_{X}$ are replaced by certain pluricanonical sheaves $\omega^{[r]}_{(X,O)}$ consisting of rational differential $r$-forms on $X$ which have constant residues over each codimension one non-normal point of $X$ (see Section 2). The author study weakly log canonical singularities, in particular in Section 3, he finds a criterion for $\operatorname{Spec}k[M]$ to satisfy Serre's property $S_2$. This result generalizes the one given by \textit{N. Terai} [in: Affine algebraic geometry. Dedicated to Masayoshi Miyanishi on the occasion of his retirement from Osaka University. Osaka: Osaka University Press. 449--462 (2007; Zbl 1128.13014)]. \par In this paper the author classifies toric varieties $X $ which are weakly (semi-) log canonical. The classification is combinatorial, expressed in terms of the log structure on the normalization, and certain incidence numbers of the irreducible components. \par In Section 4 he defines weakly normal log pairs, and weakly log canonical singularities. In Section 5, he finds a criterion for $\operatorname{Spec}k[M]$, endowed with a torus invariant boundary $B$, to be a weakly normal log pair. For part I, see [the author, ``On toric face rings. I'', Preprint, \url{arXiv:1705.02759}]. For the entire collection see [Zbl 1400.13003].