Math 245A Topics in algebraic geometry: Introduction to intersection theory in algebraic geometry

Lectures: Mondays 9-10:50 and Wednesdays 10-10:50 (not the times listed in the course guide) as well as Friday Oct. 1, 10-10:50.

Office hours: By appointment, in 380-383M (third floor of the math building). I'll usually be in Mondays and Wednesdays 2:15-3 (my 210A office hours). I will almost always be available to talk at length after each class, and at other times of the week as well.

Our goal: We'll develop intersection theory in algebro-geometric context, which will allow us to deal with singular spaces. In Intersection theory, Fulton develops a very powerful machine quite cleanly and quickly. He requires a reasonable comfort level with schemes. If you haven't seen them before and are ambitious, you should be able to follow the class. (This will involve chatting with me about any points you find confusing.) The technical tools are developed in the first eight or so chapters. The remaining chapters are important applications, and can be read in almost any order once the basic theory is developed. I would like to cover the first 8 chapters (including Chern classes, and intersection theory on nonsingular varieties), and then (in some order) Chapters 9 (excess and residual intersection), 14 (degeneracy loci and Grassmannians), and 15 (Riemann-Roch for nonsingular varieties); most likely we'll settle for less.

Reference: Fulton's Intersection theory (either edition). It will be very helpful if you have a copy.

Homework: Unlike most advanced graduate courses, there likely will be homework. If there is any, it won't be onerous; your main task will be to understand the subject.

Notes: Notes for many of the classes in ps and pdf formats will be posted here. They are very rough, some rougher than others! For the most part I will be following Fulton (with some additional explanation), and I make no claim of originality. (If the pdf file looks funny on your computer or printer, please let me know, so I can try to fix it!) Thanks to Justin Walker for many improvements.

  • Class 1 (Mon. Sept. 27): welcome, examples, strategy (ps, pdf).
  • Class 2 (Wed. Sept. 29): generic points, orders of rational functions along Weil divisors, rational equivalence, Chow group, proper/projective/finite morphisms (ps, pdf).
  • Class 3 (Fri. Oct. 1): proper/projective/finite morphisms, proper pushforwards of cycles (ps, pdf).
  • Class 4 (Mon. Oct. 4): proper pushforward of rational equivalence, flat morphisms, fundamental cycle, pulling back cycles and rational equivalences, proper pushforward commutes with flat pullback (ps, pdf). (Homework assigned!)
  • Class 5 (Wed. Oct. 6, taught by Rob Easton): excision exact sequence, affine bundles, Cartier divisors, pseudodivisors and associated terminology.
  • Class 6 (Mon. Oct. 11): more on divisors, intersecting with pseudodivisors, first Chern class of a line bundle, Gysin pullback to a(n effective Cartier) divisor. Homework "due". (ps, pdf).
  • Class 7 (Wed. Oct. 13): proving various consequences of key theorem of Chapter 2; crash course on blowing up (ps, pdf).
  • Class 8 (Mon. Oct. 18): finishing Chapter 2; projective bundles and O(1), Segre classes, properties of Segre classes (ps, pdf).
  • Class 9 (Wed. Oct. 20): Chern classes and their properties (ps, pdf).
  • Class 10 (Mon. Oct. 25): the splitting principle; applications of Chern classes (ps, pdf).
  • Class 11 (Wed. Oct. 27): Chow groups of vector bundles and projective bundles. (ps, pdf).
  • Class 12 (Mon. Nov. 1): cones; Segre classes of cones and subschemes. (ps, pdf).
  • Class 13 (Wed. Nov. 3): the Segre class of a subscheme behaves with respect to proper pushforwards and flat pullbacks. (ps, pdf).
  • Class 14 (Mon. Nov. 8): multiplicity of a variety along a subvariety, deformation to the normal cone, specialization to the normal cone, gysin pullback for lci's, intersection products on smooth varieties. (ps, pdf).
  • Class 15 (Wed. Nov. 10, taught by Andy Schultz): linear series. (ps, pdf).
  • Class 16 (Mon. Nov. 15): intersection products, refined Gysin homomorphisms, i shriek. (ps, pdf).
  • Class 17 (Wed. Nov. 17): i shriek again; refined Gysin pullback behaves well (commutes with proper pushforward and flat pullback and other refined Gysins), excess intersection formula and self-intersection, functoriality, local complete intersection morphisms. (ps, pdf).
  • Class 18 (Mon. Nov. 22): Grothendieck K groups, Grothendieck-Riemann-Roch, proof for projective space. (ps, pdf).
  • Class 19 (Wed. Nov. 24): Grothendieck-Riemann Roch proof for projective morphisms between smooth varieties. (ps, pdf).
  • Class 20 (Mon. Nov. 29): Bivariant intersection theory (ps, pdf).
  • Class 21 (Wed. Dec. 1): Bivariant intersection theory continued (notes included in previous day's notes).
    Back to my home page.
    Ravi Vakil
    Department of Mathematics Rm. 383M
    Stanford University
    Stanford, CA
    Phone: 650-723-7850 (but e-mail is better)
    Fax: 650-725-4066