Welcome to MATH 5251 -- Algebraic Geometry I (Fall 2025)! This is the first semester of a graduate course on algebraic geometry given at HKUST during the academic year 2025-2026. The course covers the foundations of modern algebraic geometry, focusing on the theory of varieties and schemes (roughly Chapters I and II of Hartshorne's book)
Course outline: here.
Details can be found here.
Details can be found here.
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.]
Course Description:
MATH 3131 is tailored for honor students who have completed MATH 2131: Honors in Linear and Abstract Algebra I.
This course offers an in-depth exploration into the realm of abstract algebra. Students will engage with fundamental algebraic structures including Groups, Rings, Modules, and Fields, as well as the application of these concepts.
Course Details:
Lectures: Wednesday and Friday, 13:30 – 14:50, Room 4502.
Tutorials: Monday, 19:00 – 19:50, Room 2463.
Course Resources:
Lecture Notes (continuously expanded and updated throughout the course).
Lecture Videos: Available on Canvas.
Midterm Exam: Problems and Solutions.
Final Exam: May 27th, 2024, 04:30 PM - 07:30 PM, at LG5 Multi-function Room.
In the spring of 2023, I co-lectured the undergraduate course algebraic geometry with Pavel Safronov. Based on lecture notes of Pavel, Miles Reid's book "Undergraduate Algebraic Geometry", and William Fulton's book "Algebraic curves: an introduction to algebraic geometry" .
Lecture 1. Quadric surfaces. (Written notes.)
Lecture 2. Segre embedding. An algorithm for finding lines on surfaces. (Written notes.)
A supplementary note explaining why there are ten lines passing through a line on a cubic surface.
Lecture 3. Cubic surfaces and 27 lines. (Written notes.)
A supplementary note on double sixers.
Lecture 4. Rationality of cubic surfaces. (Written notes.)
Lecture 5. Blowups and resolutions of singularities. (Written notes.)
Lecture 6. Blowups, eliminations of indeterminacy of rational maps, and Cremona involutions. (Written notes.)
In the fall of 2022, I teach the second half of the SMSTC course Groups, Rings & Modules.
Lecture 1. Introduction to Modules. (In-class written notes.)
Lecture 2. Noetherian properties and Hilbert basis theorem. (In-class written notes.)
Lecture 3. Integral domains (Euclidean domains, PIDs, UFDs). (In-class written notes.)
Lecture 4. Algebraic numbers and rings of integers. (In-class written notes.)
Lecture 5. Introduction to Algebraic Geometry. (In-class written notes.)
(Assignments and Suggested Solutions are available in the SMSTC system.)
In the spring of 2021, I co-taught the GlaMS course on Homological Algebra, organized by Ben Davison. I am in charge of the second block: examples and applications, and here are the links to the materials of this block:
Lecture 2.1. Tor and Ext functors. notes, video, and real-time written notes.
Lecture 2.2. Hochschild (co)homology. notes, video, and real-time written notes.