Simple surface singularities

TL;DR

This paper explores the classification of simple normal surface singularities, proposing a conjecture linking them to rational singularities derived from rational double and triple points, and provides partial proofs.

Contribution

It introduces a conjecture connecting simple surface singularities to rational singularities with specific resolution graph modifications and proves one direction of this conjecture.

Findings

Proves one direction of the conjecture for rational singularities.

Identifies classes of rational quadruple points and sandwiched singularities where the conjecture holds.

Suggests a characterization of simple surface singularities via resolution graph modifications.

Abstract

By the famous ADE classification rational double points are simple. Rational triple points are also simple. We conjecture that the simple normal surface singularities are exactly those rational singularities, whose resolution graph can be obtained from the graph of a rational double point or rational triple point by making (some) vertex weights more negative. For rational singularities we show one direction in general, and the other direction (simpleness) within the special classes of rational quadruple points and of sandwiched singularities.