Log-terminal smoothings of graded normal surface singularities

TL;DR

This paper proves that certain weighted homogeneous surface singularities with rational homology disk smoothings can be chosen to have log-terminal total spaces, advancing understanding of their geometric properties.

Contribution

It introduces the concept of graded discrepancy for normal graded domains and shows these smoothings can be made log-terminal, a novel result in the study of surface singularities.

Findings

Rational homology disk smoothings can be chosen to be log-terminal.

Introduces finite graded discrepancy to analyze smoothings.

Establishes $ ext{Q}$-Gorenstein property for these smoothings.

Abstract

Recent work ([18], [1]) has produced a complete list of weighted homogeneous surface singularities admitting smoothings whose Milnor fibre has only trivial rational homology (a "rational homology disk"). Though these special singularities form an unfamiliar class and are rarely even log-canonical, we prove the Theorem. A rational homology disk smoothing of a weighted homogeneous surface singularity can always be chosen so that the total space is log-terminal. In particular, this smoothing is $\Q$-Gorenstein. The key idea is to define a finite "graded discrepancy" of a normal graded domain with $\Q$-Cartier canonical divisor, and to study its behavior for a smoothing.