Math 632: Algebraic Geometry II

Professor Karen E. Smith
East Hall 3074

Tuesdays and Thursdays at 10 am in 4096 East Hall.
Office hours Mondays at 1, Thursdays at 2 and by appointment.

Math 632 is a first introduction to scheme theory for "beta level" PhD students in math. Prerequisites are Math 614 and Math 631.

The Daily Update, a summary of what was discussed in class each day, including assignments and quiz announcements.

An old supplementary write-up , on pullbacks of sheaves of modules and all that.

Problems Sets

Quizzes: Quiz 1 , Quiz 2 , Quiz 3 , Quiz 4 ,

Syllabus for Math 632: Affine schemes ("Spec"), sheaves and ringed spaces, schemes, projective schemes ("Proj"), varieties as schemes, coherent sheaves, vector bundles, invertible sheaves and divisors, properness and separatedness, cohomology of coherent sheaves (including computation using the Cech cohomology), the Riemann-Roch theorem for curves with applications. We will treat schemes as ringed spaces, but occassionally especially in examples, get exposure to the idea of a scheme as a "functor of points."

Expectations: The only way to really learn this material is by doing a great deal of hard exercises (mostly from Hartshorne): lots will be assigned every week. These will be collected every other week though only a small number will actually be graded. Indeed, you can easily find all the solutions on the internet so it is silly for me to assign grades based on homework. A weekly quiz will confirm that students are mastering the basic definitions and examples. The grading in this course will be somewhat less rigorous than in 631. Students who regularly show up and take quizzes, plus do some homework can expect at least an A-. After all, you are going to be hard at work on something, as PhD students already getting into research. Students who work hard on Math 632 and prove it to me (for example, as evidenced by class discussion or especially office hours discussion) can expect even higher grades.

Text: Shafarevich, Basic algebraic geometry (Vol II) AND Hartshorne, Algebraic Geometry.

More recommended Reading:

  • Eisenbud and Harris "Schemes: the language of modern algebraic geometry."
  • Eisenbud "Commutative Algebra with a view toward algebraic geometry."
  • Atiyah and MacDonald "Commutative algebra"