MAT 589: Introduction to Algebraic Geometry (Spring 2021)
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.
Please see the syllabus for additional information about the course, including university-wide policies.
Time and location
We meet on Tuesday and Thursday, 1:15–2:35 pm, in Physics P–127.
My office hours are Friday, 1pm–3pm, on Zoom.
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)
Schedule
Please study the assigned text carefully before class. Unless otherwise indicated, all page numbers refer to the above lecture notes by Gathmann.
| Week | Dates | Reading |
| 1 | Feb 4 | Introductory lecture (no reading) |
| 2 | Feb 9 & 11 | Affine varieties (pp. 6–11) |
| 3 | Feb 16 | Zariski topology, irreducible components (pp. 12–16) |
| 3 | Feb 18 | Dimension and codimension (pp. 17–21) |
| 4 | Feb 23 | Regular functions (pp. 22–26 top) |
| 4 | Feb 25 | Sheaves and ringed spaces (pp. 27–29) |
| 5 | Mar 2 | Morphisms (pp. 29–34) |
| 5 | Mar 4 | Prevarieties and varieties (pp. 35–41) |
| 6 | Mar 9 & 11 | Projective varieties I (pp. 42–51) |
| 7 | Mar 16 | Projective varieties II (pp. 52–59) |
| 7 | Mar 18 | Grassmannians (pp. 61–66) |
| 8 | Mar 23 | Finite morphisms (Shafarevich, pp. 60–65 and Appendix A as needed) |
| 8 | Mar 25 | Dimension of fibers (Shafarevich, pp. 75–77 and construction on p. 70) |
| 9 | Mar 30 | Birational maps and blowing up (pp. 67–72, Remark 9.25 and after) |
| 9 | Apr 1 | Tangent cones and tangent spaces (pp. 72–74 and pp. 77–79) |
| 10 | Apr 6 | Smooth varieties (pp. 79–83) |
| 10 | Apr 8 | The 27 lines on a cubic surface (pp. 84–88) |
| 11 | Apr 13 & 15 | Schemes (pp. 89–100) |
| 12 | April 20 | Sheaves of modules (pp. 101–108) |
| 12 | April 22 | Quasicoherent sheaves (pp. 119–116) |
| 13 | April 27 | Differentials (pp. 117–122) |
| 13 | April 29 | Cohomology (pp. 123–126 middle) |
| 14 | May 4 | Cohomology (pp. 126–131) |
| 14 | May 6 | Three theorems about cohomology (Artin, pp. 172–174) |
| 15 | May 10 | Bezout's theorem (Artin, pp. 175–176) |
Homework assignments
During most weeks, I will be collecting written homework; we will also talk about some problems in class. For each assignment, please write up your solutions nicely and hand them in by the due date. You can either send your homework to me by email (in PDF if possible), or hand in a printed copy at the beginning of Tuesday's class (stapled and with your name on the first page).
| Week | Due Date | Assignment |
| 3 | Feb 23 | 2.18, 2.23, 2.24, 2.33, 2.40 |
| 4 | Mar 2 | See this PDF file |
| 5 | Mar 9 | 4.12, 4.13, 5.7, 5.22, 5.23 |
| 6 | Mar 18 | 6.13, 6.28, 6.29, 6.30, 6.35 |
| 7 | Mar 25 | See this PDF file |
| 8 | Apr 1 | 9.8, 9.18, 9.21, 9.28 |
| 9 | Apr 13 | 10.13, 10.17, 10.18, 10.23, 10.24 |
| 11 | Apr 20 | 12.14, 12.15, 12.22, 12.36, 12.43 |
| 12 | Apr 27 | 13.8, 13.18, 13.20, 13.24, 13.26 |
| 14 | May 6 | Gathmann, 14.11, 14.12; Artin, 7.10.1, 7.10.15, 7.10.17 |