Math 763 -- Algebraic Geometry I -- Detailed list of topics

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Here is a detailed list of topics that I have covered in the past, based on 29 75-minute-classes. (To tell the truth, it seems very optimistic now... but who knows?)

  1. What is AG? Algebraic sets vs ideals. Statement of the Nullstellensatz.
  2. Proof of the Nullstellensatz. Zariski topology on $k^n$.
  3. Regular functions and regular maps. Coordinate rings.
  4. Zariski topology on algebraic varieties. Noetherian topological spaces. Principal open sets.
  5. Locally defined regular functions. Regularity of a function is local.
  6. Subvarieties of ${\mathbb A}^n$.
  7. Abstract algebraic varieties. Separated varieties.
  8. Subvarieties and products of varieties.
  9. Rational functions and rational maps.
  10. Dimension.
  11. Dimension of hypersurface.
  12. Complete intersections. Dimensions of fibers. ${\mathbb P}^n$.
  13. Projective varieties. Projective Nullstellensatz.
  14. Projective varieties are complete. Segre embedding.
  15. Grassmannians. Incidence variety.
  16. Dimension of fibers of projective maps.
  17. Chevalley's Theorem. Tangent space.
  18. Differential of a map. Smoothness. Local parameters.
  19. Taylor decomposition at a smooth point. Completed local ring.
  20. Regular local ring is a UFD.
  21. Smooth subvariety is lci. Birational vs biregular classification.
  22. Blow-ups.
  23. Resolution of singularities. Castelnuovo's criterion. Minimal surfaces.
  24. Divisors on smooth varieties. Weil divisors vs Cartier divisors.
  25. Pricipal divisors and the Picard group.
  26. Divisors on an affine variety as (fractional) ideals. Divisor classes as invertible modules.
  27. Algebraic vector bundles and line bundles.
  28. Linear systems and sections of line bundles.
  29. The Riemann-Roch Theorem.