MAT 589: Introduction to Algebraic Geometry (Spring 2026)
About the course
In a nutshell, algebraic geometry studies systems of polynomial equations and the geometry of their solution sets. It is one of the oldest branches of mathematics, with many connections to other areas such as number theory, complex geometry, combinatorics, or theoretical physics.
The plan is to spend a few weeks on basic algebraic geometry (affine and projective varieties and their properties) and then to discuss more advanced topics (schemes and sheaves, cohomology, flatness, differentials). I plan to assign homework and provide written solutions for all the questions. If you are an undergraduate student taking this course, your grade will be determined by your homework and by a brief oral examination at the end of the semester.
Time and location
We meet on Monday and Wednesday, 9:30–10:50 am, in Math 4–130.
My office hours are Friday, 10am–1pm (or by appointment).
Recommended texts
- Algebraic Geometry (by Andreas Gathmann)
- Notes for a Course in Algebraic Geometry (by Michael Artin)
- The Red Book of Varieties and Schemes (by David Mumford)
- Basic Algebraic Geometry I (by Igor Shafarevich)
- Ideals, Varieties, and Algorithms (by David Cox, John Little, and Donal O'Shea)
- Algebraic Geometry (by Robin Hartshorne)
Schedule
I will provide references for each lecture here. (Ga = Gathman, Sha = Shafarevich)
| Week | Date | Topics | Reference |
| 1 | Jan 26 | Snow day | Time & Date |
| Jan 28 | Introduction, affine varieties, Nullstellensatz | Ga §1 | |
| 2 | Feb 2 | Zariski's lemma, correspondence, Irreducible components | Ga §1 |
| Feb 4 | Zariski topology, dimension | Ga §2 | |
| 3 | Feb 9 | Hypersurfaces, morphisms, finite morphisms | Ga §2, Sha §5.3 |
| Feb 11 | Noether normalization, dimensions of fibers | Sha §6.3 | |
| 4 | Feb 16 | Projective space, Zariski topology | Ga §6 |
| Feb 18 | Homogeneous ideals, products of projective spaces | Ga §6–7 | |
| 5 | Feb 23 | Snow day | Time & Date |
| Feb 25 | Grassmannians | Ga §8 |
Homework assignments
During most weeks, I will be collecting written homework; this is mandatory for undergraduate students, optional for everyone else. Please hand in your solutions at the beginning of Monday's class (stapled and with your name on the first page).
| Number | Due Date | Assignment |
| 1 | Feb 16 | Gathman 1.19,1.21,1.22,2.17,2.23,2.36,2.40 |
| 2 | Feb 23 | See this PDF |
| 3 | Mar 4 | See this PDF |