Math 245: Intersection Theory

Winter 2011

Monday 9:40-10:55 and Friday 8:40-9:55 (with exceptions, announced on the email list) in 380-F

In this class, we'll discuss intersection theory in algebraic geometry, roughly using Eisenbud and Harris' draft book "3264 and all that". My intent is to get across some of the main ideas while doing lots of calculations. Please be sure you are on the course email list (by asking me).

  • Mon. Jan. 3: introduction. What we are hoping for in a theory of intersections, including "constancy of intersection number". Every degree n polynomial has n roots. A projective quasifinite morphism of smooth varieties of the same dimension is finite and flat. Statement of Bezout's theorem, including the pathology in P^4 and Serre's patch.
  • Wed. Jan. 5: The 27 lines on a cubic surface.
  • Mon. Jan. 10: more desiderata with intersection theory (Schubert classes, enumerative geometry, Weil conjectures, Kleiman's criterion).
  • Fri Jan. 14: the Grothendieck group and its associated graded groups.
  • Mon. Jan. 17: no class (MLK day).
  • Fri. Jan. 21: intersection numbers: intersecting line bundles with a coherent sheaf (or a subvariety).
  • Mon. Jan. 24: polynomiality, Riemann-Roch, and asymptotic Riemann-Roch.
  • Fri. Jan. 28: the Hodge index theorem.
  • Mon. Jan. 31: global generation / ampleness / very ampleness. Statement of characterizations of ampleness, and start of proof.
  • Fri. Feb. 4: restatement of ampleness for proper schemes over an affine. Proof of equivalences (some with Noetherian hypotheses). Generalizations of notions of ampleness, very ampleness, and global generation.
  • Mon. Feb. 7: the Nakai-Moishezon criterion, and proof. Nef divisors: definition, first properties, examples.
  • Fri. Feb. 11: Kleiman's theorem, and Kleiman's criterion for ampleness.
  • Mon. Feb. 14: "Chow groups" as associated graded groups of the Grothendieck group of coherent sheaves; properties and examples.
  • Fri. Feb. 18 cancelled due to illness.
  • Fri. Feb. 25: Segre and Chern classes and their properties.
  • Mon. Feb. 28: more on Chern classes, including applications.
  • Comparison of Grothendieck groups of vector bundles and coherent sheaves. The "Chow ring" of a regular variety.
  • Fri Mar. 4: K(projective space). The Chern character, the Todd class, and the statement of Hirzebruch-Riemann-Roch. Examples. Proof for projective space over a field. Grothendieck-Riemann-Roch: statement and philosophy. Advertisement for next week: if we know GRR for P^n_Y rightarrow Y and for closed immersions, we know HRR in general, and GRR in many circumstances.
  • Mon. Mar. 7: more GRR discussion. GRR for P^n_Y rightarrow Y. Reduction of the proper case to the projective case. Reduction of the closed immersion case via deformation to the normal cone.
  • Fri. Mar. 11: Conclusion of proof of GRR. Summary of course: what we learned.
  • Here is a wonderful image of Grothendieck-Riemann-Roch, in the master's hand:
    Grothendieck in the Witches Kitchen

    (I'd mentioned its existence, and Mike Lipnowski managed to track it down.) An alleged translation:

    "Witches Kitchen 1971. Riemann-Roch Theorem: The final cry: The diagram is commutative! To give an approximate sense to the statement about f: X → Y, I had to abuse the listeners' patience for almost two hours. Black on white (in Springer lecture notes) it probably takes about 400, 500 pages. A gripping example of how our thirst for knowledge and discovery indulges itself more and more in a logical delirium far removed from life, while life itself is going to Hell in a thousand ways and is under the threat of final extermination. High time to change our course!"

    Thanks to an anonymous helper for making the picture the right size!

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