Research Notes on Resolution Graphs and Plane Curve Singularities

Research Notes on Resolution Graphs and Plane Curve Singularities

From Mathematical Lectures to Computational Research

How can the singularity of an algebraic curve be transformed into a geometric structure and represented by a combinatorial graph?

This book explores the relationship between algebraic geometry and topology through the study of plane curve singularities, blow-ups, embedded resolutions, and weighted dual resolution graphs.

Prepared as part of the Budapest REU Mathematics Program 2026 in Budapest, Hungary, these research notes bring together expanded and reorganized lecture materials, detailed mathematical explanations, worked examples, geometric visualizations, and computational exploration using SageMath.

The book develops mathematical ideas step by step, from the complex projective line and the geometry of blowing up points to the construction and interpretation of resolution graphs.

Particular attention is given to understanding how local algebraic equations can reveal geometric and topological information about singular curves.

The notes also explore connections with Puiseux data, intersection multiplicities, the Milnor fibre, monodromy, and mathematical invariants.

Designed for undergraduate mathematics students, beginning graduate students, and readers interested in algebraic geometry and singularity theory, this book provides a structured introduction to the mathematical ideas underlying the resolution of plane curve singularities.

Author: Saran Phosuwan
Budapest REU Mathematics Program 2026
Budapest, Hungary