Algebraic Geometry
The main objective of the course is to introduce students to local algebraic geometry, with a focus on plane curve singularities. It aims to provide insight into the singularity theory of plane curves and the geometric theory of valuations of the ring of convergent series of two variables over the complex numbers. The course will demonstrate that singular points of algebraic curves in the complex plane are a meeting point for various areas of mathematics, including algebra, topology, and complex analysis.
Through this exploration, students will gain a deep understanding of two powerful tools for the resolution of singularities: parametrization via Puiseux series and suitable change of variables via blowing-ups. With these techniques, the course offers a deep dive into the topological and analytic classification of singularities of plane curves, and some related invariants.
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