It was conjectured by McKernan and Shokurov that for any Fano contraction $f:X \to Z$ of relative dimension r with X being $\epsilon $-lc, there is a positive $\delta $ depending only on $r,\epsilon $ such that Z is $\delta $-lc and the multiplicity of the fiber of f over a codimension one point of Z is bounded from above by $1/\delta $. Recently, this conjecture was confirmed by Birkar [9]. In this article, we give an explicit value for $\delta $ in terms of $\epsilon ,r$ in the toric case, which belongs to $O(\epsilon ^{2^r})$ as $\epsilon \rightarrow 0$. The order $O(\epsilon ^{2^r})$ is optimal in some sense.

We prove that the canonical ring of a canonical variety in the sense of de Fernex and Hacon is finitely generated. We prove that canonical varieties are Kawamata log terminal (klt) if and only if is finitely generated. We introduce a notion of nefness for non-ℚ-Gorenstein varieties and study some of its properties. We then focus on these properties for non-ℚ-Gorenstein toric varieties.

In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being ℚ-Gorenstein or the pair being log ℚ-Gorenstein. The main features of the theory extend to this setting in a natural way.