Course Description:
MATH 5261 is the second part of a two-part graduate course in Algebraic Geometry. This semester, we will focus on using cohomological tools to study algebraic geometry.
Meeting Time and Venue: Weekly on Mondays, 15:00 -18:00, CYTG001.
Course Outline: Here.
Lecture 1 (Feb 3): Overview and Prelude to Derived Functors.
Lecture 2 (Feb 10): Derived Functors.
Lecture 3 (Feb 17): Sheaf Cohomology and Derived Categories I.
Lecture 4 (Feb 24): Derived Categories II, and Spectral Sequences.
Lecture 5 (Mar 3): Čech Cohomology.
Lecture 6 (Mar 10): Serre‘s Theorems, Finiteness, and Affine Criterion.
Lecture 7 (Mar 17): Ample Line Bundles.
Lecture 8 (Mar 24): Serre Duality.
Lecture 8.5-9.5 (Mar 31): Proof of Serre Duality. Flat Modules and Algebras. (Notes are integrated into previous and subsequent lecture notes.)
Lecture 9. (Apr 7): Flat Morphisms and Faithfully Flat Descent.
Lecture 10 (Apr 14): Base-change and Semi-continuity.
Lecture 11 (Apr 28): Smooth Morphisms.
Extensive course materials, including lecture notes, readings, and assignments, are available on the course’s Canvas pages.
Course Description:
MATH 5251 is the first part of a two-part graduate course in Algebraic Geometry at HKUST for the academic year 2024-2025. This semester, we will focus on the theory of varieties and schemes.
Meeting Time and Venue: Classes are held weekly on Thursdays from 3:00 PM to 6:00 PM in Room 5564 (Lift 27-28).
Lecture Notes:
Section 1. Affine Varieties
Section 2. Projective Varieties
Section 3. Sheaves
Section 4. Affine Schemes
Section 5. Schemes
Section 6. Basic Properties of Schemes
Section 7. Functor of Points. Fibre Products.
Section 8. Separated and Proper Morphisms.
Section 9. Sheaves and Bundles.
Section 10*. Projective Morphisms.
[Note: Section 10 is optional and only for students' reference. The Proj theory will not be in the exam and is not required for subsequent topics. The theory of ample line bundles will be revisited next semester.]