Discrepancies of normal varieties

We give an example of a non Q-Gorenstein variety whose canonical divisor has an irrational valuation and an example of a non Q-Gorenstein variety which is canonical but not klt. We also give an example of an irrational jumping number and we prove that there are no accumulation points for the jumping numbers of normal non-Q-Gorenstein varieties with isolated singularities. We prove that the canonical ring of a canonical variety in the sense of [dFH09] is finitely generated. We prove that canonical varieties are klt if and only if R(−KX) is finitely gener-ated. We introduce a notion of nefness for non-Q-Gorenstein varieties and study some of its properties. We then focus on the properties of non-Q-Gorenstein toric varieties, with particular attention to minimal log discrepancies.