Math 256B - Algebraic Geometry

InstructorPaul Vojta
LecturesMWF 1:10–2:00, online
Class Number 22869
Office883 Evans
Office HoursTBD
PrerequisitesMath 256A
Required TextHartshorne, Algebraic Geometry, Springer
Catalog Description Math 256AB: Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall.
SyllabusWe will do much of Chapter III (Cohomology), followed by some material on smoothness of morphisms.
GradingGrades will be based on homework assignments. There will be no final exam, but the last problem set will be due about a week after the last day of classes.
Homework Weekly or biweekly, assigned in class
CommentsI tend to follow the book fairly closely, but try to give interesting exercises and examples.


No.DateTitle Download
1March 1 A lemma in the solution of Hartshorne III, Ex. 4.4 dvi pdf
2April 6 A lemma for duality dvi pdf
3April 8 Spectral sequences dvi pdf
4April 19 Associated primes dvi pdf
5April 25 Some local algebra for II 8.17 dvi pdf
6April 25 Proof of II 8.17 dvi pdf
7April 27 A lemma on smoothness [superseded by Handout 9] dvi pdf
8May 2 A lemma for (III, Ex. 10.3) (étale implies unramified) dvi pdf
9May 6 The Jacobian criterion for smoothness dvi pdf

Homework Assignments

Solutions will be posted on bCourses after each assignment has been graded.

To submit your assignments, use bCourses. I have set it to accept only pdf files.

12/1 dvi pdf
22/8 dvi pdf
32/16 Hartshorne III Ex. 2.3, 3.1, 3.2 Ex. 3.2 is (NC)
42/24 Hartshorne III Ex. 4.1, 4.3, 4.5 Ex. 4.1 is (NC)
53/7 dvi pdf
63/15 Hartshorne III Ex. 5.8, 5.10, 6.3 Ex. 5.10 and 6.3 are (NC)
73/31 dvi pdf
84/15 dvi pdf
94/28 dvi pdf
105/14 dvi pdf

The following policies apply to the homework assignments.

  1. Discussion and the exchange of ideas are essential to doing academic work. For assignments in this course, you are encouraged to consult with your classmates as you work on problem sets. However, after discussions with peers, make sure that you can work through the problem yourself and ensure that any answers you submit for evaluation are the result of your own efforts. In addition, you must cite any books, articles, websites, lectures, etc. that have helped you with your work using appropriate citation practices (other than class lectures and corresponding parts of the textbook). Similarly, you must list the names of students with whom you have collaborated on problem sets. (This paragraph was adapted from, accessed on 28 January 2013.)
  2. As an exception to the above, you may not collaborate on problems marked “(nc).”
  3. For problems from the book, you may use without proof any lower-numbered exercises from the same section or any exercises from sections covered earlier. For other exercises, you may use without proof any exercises from earlier sections in the book; for the current section(s), ask. For any exercises, if you weren't able to do an exercise or part of an exercise, you may still assume that it holds for subsequent exercises or parts.
  4. Credit may be reduced if you look up the answer in some other source (as opposed to using the material from elsewhere to aid in your understanding), or if you look ahead in the textbook.

Last updated 7 May 2021