A combinatorial approach to the classification of resolution graphs of weighted homogeneous plane curve singularities

Year 2018, Volume: 47 Issue: 4, 805 - 812, 01.08.2018

Muhammad Ahsan Binyamin , Hafız Muhammad Afzal Siddiqui , Amir Shehzad

Abstract

In this article we describe the classification of the resolution graphs of weighted homogeneous plane curve singularities in terms of their weights by using the concepts of graph theory and combinatorics. The classification shows that the resolution graph of a weighted homogeneous plane curve singularity is always a caterpillar.

Keywords

Plane Curves, Weights, Caterpillar

References

• Arnold, V. I.; Gusein-Zade, S. M.; Varchenko,A. N. Singularities of Differentiable Maps, Volume I, Birkhäuser, Boston Basel Berlin (1985).
• Cutkosky, S. D. and Srinivasan, H.; The algebraic fundamental group of a curve singularity, Journal of Algebra 230, 101-126, (2000).
• De Jong, T. and Pster, G.; Local Analytic Geometry, Vieweg (2000).
• Jingen, Y.; Curve Singularities and Graphs, Acta Mathematica Sinica, 6 (1), 87-96, (1990).
• Kollár, J.; Lectures on Resolution of Singularities, Princeton University Press (2007).
• Kang, C.; Analytic Types of Plane Curve Singularities defined by Weighted Homogeneous Polynomials, Trans. A.M.S. 352 (9), 3995-4006, (2000).
• Muhly, H.T. and Zariski, O.; The Resolution of Singularities of an Algebraic curve, Amer.J.Math., 61 (1), 107-114, (1939).
• Saito, K.; Quasihomogene isolierte singularitäten von hyperächen, Invent. Math. 14, 123- 142, (1971).
• Wall, C.T.C.; Singular Points of Plane Curves, Cambridge University Press (2004).