Math 8211

List of Topics of Presentations

(permanently under construction)

As you all know, each of you should make a topic presentation during the school year.

Below is a list of suggested topics for both semesters. Sections refer to Eisenbud's textbook, unless a different source is mentioned. If you find a source other than the one I suggest, you are welcome to use it. The topics below have different length and level of difficulty. I have tried to order the list by difficulty (from my own point of view:-). Keep in mind, that you have to make a 50 minute presentation, similar to a talk in a research seminar. You will have to choose only the most important/interesting points of your topic at your discretion and briefly sketch some proofs of main results or ideas beyond them and place all this within 50 minutes.

Choose a topic and tell me, so that I will mark it unavailable and discuss the date of your presentation with you. You may also choose a topic not from this list - talk to me about that.

  1. Completions (E: Chapter 7; AM: Chapter 10)
  2. Graded rings and modules (AM: Chapter 10)
  3. Monomial ideals (E: Chapter 15) Claimed by Hyeung-joon Kim for December 8
  4. Groebner bases (E: Chapter 15) Claimed by Hazem Hamdan for December 10
  5. Kaehler differentials (E: Chapter 16) Claimed by Jihoon Ryoo for the Spring term
  6. "Pathological" counterexamples of Akizuki (Nagata, Local Rings: Appendix; get a simplified version from me)
  7. Stanley-Reisner rings (Ask me for references)
  8. Valuative criteria of separatedness and properness (Mumford, The Red Book; Eisenbud-Harriss, Schemes, Chapter III A; Hartschorne, Algebraic Geometry, Chapter II)
  9. Local cohomology (E: Appendix 4) Claimed by Wenliang Zhang for the Spring Term
  10. Primary decomposition and generalized cohomology theories; Landweber's theorem (Landweber's original paper)
  11. The local Riemann-Roch theorem (papers by Feigin, Tsygan, Nest, Felder, and Shoikhet)
  12. Green-Lazarsfeld's gonality conjecture: relationship between the Koszul complex and algebraic geometry (M. Green, Koszul cohomology and the geometry of projective varieties, J. Diff. Geom. 19 (1984) 125-171; and M. Green and R. Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986) 73-90)