The purpose of this series of lectures is to present an EASY, EASY introduction to the theory of resolution of singularities over a field of characteristic zero, and then to continue onto the discussion of our approach to the problem in positive characteristic, which to this day remains open. My main goal is to make the lectures accessible to the beginning graduate students and even to the advanced undergraduate students as much as possible. The background is algebraic geometry at the level of the textbook by Hartshorne† is preferred, but not absolutely necessary. I will review some basic materials on the way.
- What is it to say a variety is nonsingular/singular?
- The very formulation of the problem of resolution of singularities.
- The definition of the "order (multiplicity)" of an ideal, which is the only invariant we use to measure the singularities.
- Resolution of singularities of a curve embedded in a nonsingular surface
- Higher dimensional Case (Inductive Scheme)
- Our approach to the problem in positive characteristic:
† Algebraic Geometry by R. Hartshorne, GTM series 52, Springer-Verlag