18.726: Algebraic Geometry

This is the home page for the course 18.726 (Algebraic Geometry), taught at MIT during the spring 2009 semester.

The class is the second in a two-semester graduate-level sequence in algebraic geometry beginning with 18.725. This sequence is intended for students planning research either in algebraic geometry or in a neighboring area in which algebraic geometry plays a major role (algebraic topology, representation theory, arithmetic geometry, etc.). While 18.725 presents a range of topics couched in the language of "classical" algebraic geometry (i.e., algebraic varieties over an algebraically closed field, or even specifically over the complex numbers), 18.726 introduces the language of "modern" algebraic geometry, including sheaves, schemes, morphisms, and cohomology.

Fast facts:

Lectures: MWF 10-11 in 2-102 by Kiran Kedlaya.
Office hours: Thurs 2:30-3:30 in 2-165 (unless I'm out of town), or by appointment.
Problem sets: to be posted below each Friday (or shortly before), to be submitted the following Friday. Electronic files can be submitted directly to our grader, Fucheng Tan (tanfch at math).

Also available:

Syllabus
List of topics, and lecture notes
Virtual office hours
Problem sets:
- Problem Set 1 (due February 13). Note: I corrected "additive category" to "preadditive category" in problem 3, but this has no effect on solving. More seriously, 3(c) was misstated; please download again (you may need to flush your browser cache first).
- Problem Set 2 (due February 20).
- Problem Set 3 (due February 27). Corrections (reflected in the file now): In problem 2, the ring should be F_2^S rather than F_2[S]. I changed 7(a) because my originally intended solution doesn't work. In problem 18, M' should be the global sections of the sheafification of the presheaf-on-a-basis \tilde{M} (since we are still trying to prove that the latter is a sheaf). Problem 20 has been corrected (again): see new posted file.
- Problem Set 4 (due March 6). Note: you may use Hartshorne's definition of projective morphisms instead of my more inclusive notion. Warning: The answer suggested in the formulation of Eisenbud-Harris II-20 appears to be incorrect.
- Problem Set 5 (due March 13). Correction: Eisenbud-Harris III-43(a) is incorrect (it contradicts the preceding theorem, which is correct). However, part (b) still makes sense, so just submit that. (Remember that this is only a required exercise if you happen to have access to Eisenbud-Harris. But I suspect this includes most of you.)
- Problem Set 6 (due March 20). Corrections: the statement of problem 3 was unclear, so I reworded it. In 7(b), I should have said "proper" instead of "of finite type".
- Problem Set 7 (due April 3). This set should be a bit less painful than the last couple, so you can relax over spring break. Corrections: there was a serious error in the statement of the embedding criterion on the divisors handout, and the canonical embedding theorem on the Riemann-Roch handout; please check those (or Hartshorne) for the correct statements before attempting the problems. Amendment: on problem 8, you may assume that k has characteristic 0.
- Problem Set 8 (due April 10). Correction: in problem 11, I said "Galois cohomology" when I meant "group cohomology".
- Problem Set 9 (due April 17). Correction: I decided to replace Problem 1 with something simpler. The replaced problem will appear in modified form on PS 10. I also changed the hint (as of Wednesday right before class). Modification: on Hartshorne Problem III.2.7, if you can't figure out how to do it directly from the definition, go ahead and use Cech cohomology instead. Modification: if you can't solve Problem 1 as written, do Hartshorne III.4.4 instead. I've prepared a supplemental handout explaining spectral sequences and how they can be used here.
- Problem Set 10 (due April 24).
- Problem Set 11. Corrections: I had the wrong definition of coherence in lecture. See Problem 5(a) for the correct definition. Problem 5(b) was misstated; it should state that if any two of F, F_1, F_2 are coherent, then so is the third. The hypotheses in Problem 7 should have included the assumption that not only is the ring A coherent, but so is A[x_1, ..., x_n] for each positive integer n. (Somewhat surprisingly, it is possible to have A coherent and A[x] noncoherent!)
- Problem Set 12. Corrections: in Problem 1 (referring to ps 8, problem 9), Ext(M,N) should have been Ext(N, M). PS 8 is now corrected. In problem 12, s_b should have been ds_b in the statements of parts (a), (b), (c).