Richard Borcherds: Algebraic Geometry

5th May, 2020

This is a 51-lecture course, with each lecture being about 30 minutes or so, given online by Richard Borcherds. It gives an introduction to algebraic geometry. It follows Chapter I of Hartshorne’s book.

  1. Introduction
  2. Two cubic curves
  3. Bezout, Pappus, Pascal
  4. Kakeya sets
  5. Affine space and the Zariski topology
  6. Noetherian spaces
  7. Weak Nullstellensatz
  8. Strong Nullstellensatz
  9. The Lasker-Noether theorem
  10. The proof of the Lasker-Noether theorem
  11. Quotients of varieties by groups
  12. Hilbert’s finiteness theorem
  13. Three examples of quotients
  14. Dimension
  15. Projective space
  16. Desargues’s theorem
  17. Affine and projective varieties
  18. Products of varieties
  19. The Veronese surface and the variety of lines in space
  20. Grassmannians
  21. Projective space bundles
  22. Toric varieties
  23. Categories
  24. Regular functions
  25. Morphisms of varieties
  26. Affine algebraic sets and commutative rings
  27. The twisted cubic
  28. Products of projective varieties
  29. Automorphisms of space
  30. The Ax-Grothendieck theorem
  31. Rational maps
  32. Elliptic functions and cubic curves
  33. Rationality of cubic surfaces
  34. Blowing up a point
  35. More on blowups
  36. The Atiyah flop
  37. Singular points
  38. The Zariski tangent space
  39. Du Val singularities
  40. Examples of resolutions
  41. Completions
  42. Resultants
  43. Proper maps
  44. Survey of curves
  45. Hurwitz curves
  46. Examples of Hurwitz curves
  47. Resolution of curve singularities
  48. Newton’s rotating ruler
  49. Hilbert polynomials
  50. The degree opf a projective variety
  51. Bazout’s theorem