Normal surface singularities of small degrees

TL;DR

This paper studies normal surface singularities of degree two, deriving formulas for their invariants and classifying their weighted dual graphs, extending the Yau sequence concept.

Contribution

It introduces a method to compute the canonical cycle and genus bounds for degree two surface singularities using the Yau cycle.

Findings

Derived the canonical cycle for certain degree two singularities.

Established formulas for arithmetic and geometric genera.

Provided classification properties for weighted dual graphs.

Abstract

The notion of the Yau sequence was introduced by Tomaru, as an attempt to extend Yau's elliptic sequence for (weakly) elliptic singularities to normal surface singularities of higher fundamental genera. In this paper, we obtain the canonical cycle using the Yau cycle for certain surface singularities of degree two. Furthermore, we obtain a formula of arithmetic genera and an upper bound of geometric genera for these singularities. We also give some properties about the classification of weighted dual graphs of certain surface singularities of degree two.