Discrepancies of non-$\Q$-Gorenstein varieties

TL;DR

This paper presents examples of non-$\Q$-Gorenstein varieties with unique canonical properties, irrational valuations, and jumping numbers, and establishes the absence of accumulation points for these jumping numbers in certain cases.

Contribution

It provides the first known examples of non-$\Q$-Gorenstein varieties with irrational canonical valuations and jumping numbers, and proves key properties about their jumping numbers.

Findings

Existence of non-$\Q$-Gorenstein varieties with irrational valuations.

Existence of irrational jumping numbers.

No accumulation points for jumping numbers in certain varieties.

Abstract

We give an example of a non $\Q$-Gorenstein variety which is canonical but not klt, and whose canonical divisor has an irrational valuation. 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.