Abstract:
The aim of this course is to provide a local study of the singular points of plane curves. The theory of singularities of complex algebraic plane curves is situated at the crossroads of many interesting areas of mathematics, making the study of curve singularities particularly fruitful up to this day.
We start with a short introduction to the subject, where we briefly review results about manifolds and plane algebraic curves. Moreover, we focus on polar curves, Puiseux's theorem and resolution of singularities. Next, we show how combinatorial tools such as the Eggers tree can be used in the study of the contact of two branches of a curve. One of the core concepts in the first part of the course is equisingularity of curves.
In the second part of the course we study the geometry of the link of a singularity, Milnor's fibration theorem and the Milnor number. In addition, we also see a proof of Klein's equation (using Euler characteristics of constructible functions), which relates invariants of a projective curve with invariants of its dual curve. Time permitting, we will also tackle the decomposition of the link complement and the computation of the monodromy of the Milnor fibration.
Bibliography:
Wall, C. T. C. Singular points of plane curves. London Mathematical Society Student Texts, 63. Cambridge University Press, Cambridge, 2004. xii+370 pp.