Manwai's webpage

Welcome to MATH 232B: Theory of scheme

Meeting Time: MW 10:30am - 11:45am @ SC 411
switch to WF 10:30am - 11:45am @ SC 411 from March 2

Major reference:

Note by Andreas Gathmann

Other reference:

The Geometry of Schemes by David Eisenbud and Joe Harris
Basic Algebraic Geometry 2: Schemes and Complex Manifolds by Igor R. Shafarevich
Algebraic Geometry by Robin Hartshorne
The Rising Sea: Foundations of Algebraic Geometry by Vakil

Contact information:

Man Wai (Mandy) Cheung @ Science Center room 505H
Office hour: Tue 9-10am; Thurs 9-10am
Send me an email to schedule a meeting if you cannot make these office hours.

Homework: There will be a homework assignment biweekly. One week extension is accepted only with sensible reasons. Further delay would only be accepted with official reaons. If not, one subgrade will be off for each day of delay of submittion.

Problem set 1: Hartshorne II. 1.3, 1.17, 1.21 + Question 1-4 in PS 1(due on Feb 12)
Problem set 2: Hartshorne II. 2.3, 2.5, 2.7, 2.13 + E & H Exercise II-12 + III-6 (due on Feb 26)
Problem set 3: Hartshorne II. 3.7, 3.8, 3.10, 3.11 + E & H Exercise II-25 (due on Mar 11)
Problem set 4: Hartshorne I. 6.2; II. 4.1, 4.7, 4.12, 6.1 (due on Apr 1)
Problem set 5: Hartshorne II. 5.3, 5.8, 6.2, 6.6 (due on Apr 15)
Problem set 6: Hartshorne II. 5.13, 5.14, 7.7, 8.4, 8.5, 8.6 (optional)

Mid-term project (due on Mar 27): 1 page description of the final project.
Final project (due on May 4): a survey article of a topic related to the course materials.
Grading: The final grade in the course will be based on the homework grades and a final project. The final project would be take-home writing project/ exam / presentation.

Schedule of the class:

Jan 27 -- Presheaves, sheaves
Jan 29 -- Ringed space, prevarieties, gluing of prevarieties, Spec A
Feb 3 -- locally ringed space, affine scheme
Feb 5 -- k-valued point, reduced scheme, examples of non-reduced scheme, multiplicities
Feb 7 -- gluing of schemes, projective scheme
Feb 10 -- irreducible, integral, reduced scheme over algebraic closed field, Noetherian
Feb 12 -- morphism finite type, closed immersion, closed subscheme, embedded point
Feb 19 -- fibre product
Feb 21 -- flat family of schemes, universal hypersurfaces
Feb 26 -- separated and proper morphisms
Mar 4 -- separated and proper morphisms (continue)
Mar 11 -- divisors
Spring break
Mar 25 -- divisors (continues), divisors on curves
Mar 27 -- Cartier divisors
Apr 1 -- elliptic curves, Invertible sheaves
Apr 3 -- Picard group, sheaves of modules
Apr 8 -- Quasi-coherent sheaves, vector bundles
Apr 10 -- Projective morphisms
Apr 15 -- Ampleness, linear systems
Apr 17 -- Differentials
Apr 22 -- Curves, Riemann-Roch
Apr 24 -- Surfaces
(Apr 27) -- Monomial transformation
Apr 29 -- Cubic surface (27 lines!)