
Math 2050: Algebraic geometry, Fall 2017
MWF 1010:50 Kassar House 105
Instructor: Melody Chan
office: Kassar House 311
course email: melody_chan+2050@brown.edu
(please use this email address for courserelated emails)
course website: http://www.math.brown.edu/~mtchan/2017Fall_2050.html
office hours, Mondays 34, Wednesdays 45 (for now), and by appointment
Course description and goals
This is the first semester of a yearlong graduate course in algebraic geometry. We will cover the foundations of scheme theory: affine and projective schemes, various properties thereof; dimension, morphisms of schemes, fibered product and base change, and more as time permits. The main textbook for this course is Qing Liu's Algebraic geometry and arithmetic curves, 2006 paperback edition. We will cover as much of Chapters 24 as time permits.
The prerequisites are commutative algebra at the level of Math 25102520, including familiarity with rings and modules, tensor product and localization, various finiteness conditions, flatness... Some knowledge of general topology is also necessary, and a basic familiarity with manifolds will also be very helpful for understanding what is going on. If you have any questions about prerequisites, please let me know.
Required textbook
Qing Liu, Algebraic geometry and arithmetic curves, 2006 paperback edition.
This will be the main textbook for the course.
This book is available at the bookstore for $85 new, $63.75 used.
Please note also that with a Brown University login you can read the current edition of the book online. More conveniently,
individual chapters of the previous 2002 edition may be downloaded in PDF.
The author maintains a list of errata here.
Supplementary textbooks
David Eisenbud and Joe Harris, Geometry of schemes (available online). This is a great book for some supplementary examples, exercises, and intuition. I will probably assign problems from both Liu and EisenbudHarris.
Ravi Vakil, The rising sea: Foundations of algebraic geoemtry (available online). This is a great learnityourself pathway into the subject, full of exercises to work out.
Grading
Weekly problem sets will be posted here, typically due once a week on Fridays, at the beginning of class in hard copy (LaTeX strongly preferred) and stapled. No late problem sets will be accepted. However, your lowest score will be dropped. The final grade will be based entirely on the problem sets.
All problem sets in one PDF
Schedule
Week 1: Introduction. Spec of a ring, Zariski topology.
Week 2: Algebraic sets. Finiteness, Noether normalization. Nullstellensatz. Definition of presheaves and sheaves.
Week 3: Sheaves, stalks, exact sequences. Sheaves on a base. Morphisms of sheaves, pushforward and pullback. Ringed topological spaces. Definition of affine scheme.
Week 4: Schemes. Morphisms of schemes. Open and closed immersions. Spec and global sections are adjoint. Gluing schemes. Examples. Projective space.
Week 5: Schemes over a base. Functor of points. Construction of fiber products. Fibers of morphisms; base change. Examples.
Week 6: Reduced, irreducible schemes. Noetherian conditions. Examples: nonreduced points.
Week 7: Primary decomposition. Flat families over a 1dimensional base. Integral schemes. Examples: local schemes.
Week 8: Dimension. Projective varieties; Proj of a graded ring.
Week 9: Examples: plane curves, quadric hypersurfaces, Veronese embeddings. Grassmannians. Projective morphisms with arbitrary base; examples. Quasicompact, finite type morphisms. Affine communication. Separated morphisms.
Week 10: Proper morphisms. Examples. Valuation rings. Statement of valuative criterion. Projective implies proper: elimination theory. Examples: resultants, discriminants.
Week 11: Extending the base field; geometric reducedness/irreducibility/etc. Examples: the real affine plane. Kvalued points.
Week 12: The Frobenius map. Regularity. Thanksgiving.
Week 13: Tangent spaces and regularity; the Jacobian criterion. Normal schemes, regularity in codimension 1. Normalization.
Week 14: Student talks.
Collaboration
You are encouraged to collaborate with other students in the class on your homework, although I suggest that you think carefully about each problem on your own first. You are required to write up your solutions separately and write the names of the students with whom you worked on the assignment. (You may only use the Internet as a general reference, at the level of generality of Wikipedia.)
How much time will this class take?
Roughly speaking, you should expect to spend twelve hours every week outside of class, including attending office hours, reviewing class material and doing problem sets. In addition to three hours of class every week, I estimate a total of 15*13 = 195 hours of time spent on this class.
Accommodations for students with disabilities
Please contact me as early in the semester as possible so that we may arrange reasonable accommodations for a disability. As part of this process, please be in touch with Student and Employee Accessibility Services by calling 4018639588 or online at
http://brown.edu/Student_Services/Office_of_Student_Life/seas/index.html
