Introduction to Algebraic Geometry
We will develop the theory of algebraic varieties, which are zero sets of polynomial equations. We will start with some basic commutative algebra - define Groebner basis, chain conditions and talk about the ideal membership problem. Then we will discuss varieties - affine, projective, quasi-projective. We will prove Hilbert's Nullstellensatz (one of the most important theorems of classical algebraic geometry). We will talk about different notion of dimension and how those relate. We will define maps between varieites - morphisms, rational and birational maps. The remaining topics are (but not limited to) singularity theory, normalization and blow-ups, elimination theory, resultants, divisors on algebraic varieties. We will try to also focus on some computational aspects of algebraic geometry.
Course info
- Lectures: TTh 11:35-12:45
- Classroom: DL 431
- Office: DL 406
Syllabus
You can find the syllabus here.Office hours
If you need help with the course or you have questions regarding the material or the homeworks or you would just want to pick up your homework, please don't hesitate to stop by my office or write me an email. If the time is inconvenient you can arrange an appointment.Office hours: Monday 5-6pm
Homework Sets
- Homework Set 1 - due 13 September.
- Homework Set 2 - due 2 October.
- Homework Set 3 - due 23 October.
- Homework Set 4 - due 4 December.
Some Macaulay2 code
- Examples: Groebner basis and division algorithm.
Schedule
| Week | Dates | Topics |
|---|---|---|
| 1 | 29 Aug - 1 Sept | Introduction, monomial orders, division algorithm, introducing the ideal membership problem |
| 2 | 2 Sept - 8 Sept | Dickson's lemma, Hilbert's basis theorem, existence of Groebner basis, Noether's proposition, Buchberger's criterion |
| 3 | 9 Sept - 15 Sept | Buchberger's algorithm, examples, Macaulay2, varieties |
| 4 | 16 Sept - 22 Sept | Hilbert's Nullstellensatz | 5 | 23 Sept - 29 Sept | Coordinate rings, dimension, morphisms | 6 | 30 Sept - 6 Oct | Rational and birational maps, Projective varieties | 7 | 7 Oct - 13 Oct | Quasi-projective varieties, products of quasi-projective varieties | 8 | 14 Oct - 20 Oct | The image of projective morphism is closed | 9 | 21 Oct - 28 Oct | Families of varieties, lines on surfaces | 10 | 29 Oct - 3 Nov | Elimination theory, resultants | 11 | 4 Nov - 10 Nov | Tangent space | 12 | 11 Nov - 17 Nov | Resolution of singulatities overview, curves, normalization | 13 | 24 Nov - 1 Dec | Blow-ups |