MA8203 Algebraic Geometry (Spring 2021)

Announcements / notices

  • 7 May: updated the exam prep sheet.
  • 28 April: Sign up for a final exam time! Please sign up for a final exam time slot at the following link. Choose only one time that works for you, and be sure to enter your full name. For more information about the exam, please see under "Exam" below. The deadline for signing up is Wednesday 5 May. Link to sign up for final exam: https://doodle.com/poll/tnu5dkkasp7webui?utm_source=poll&utm_medium=link
  • 21 April: I have posted a PDF with details on the content of the final exam (see under "Exam" below), and also updated the exercise sheet to reflect the problems that may be on the final exam.
  • 12 April: We reconvene on Zoom today. Our plan is to finally prove Bezout's theorem this week. I will also discuss the final exam in class.
  • 6 April: Remember, we are not meeting on Zoom today. Instead, use the time to work on the Chapter IV and V exercises. These have been added to the exercises file below. Note also that I have added some details regarding the final oral exam for this course (see below) in terms of dates and format. We will also discuss it more in class next week.
  • 26 March: Note the change on the schedule that Tuesday, 6 April, will now be an independent work day (we will not meet on Zoom that day). The exercise list has been updated below; use the class time on that day to work on Chapter IV and V exercises on the exercise list. Also, an unrelated announcement: I am interested to know what aspects of algebraic geometry you might be pursuing now or after this course is over (this may impact what types of topics we study during the remainder of the course). If you would like to let me know what you're reading or plan to read (in algebraic geometry), please do so here: https://forms.gle/2ZEAXNq96ZJREcB36 Of course, we will probably not go into too many specifics about other topics, but it is helpful for me to know what aspects of algebraic geometry you are most interested in learning more about. Thanks!
  • 3 March: I have updated the PDF file with exercises. Please work the Chapter III exercises before class on Monday 15 March.
  • 1 March: I have created an overleaf file for you to add/edit solutions to the exercises. The link below should give you access to edit. Let me know if you have questions.

Lecturer

Schedule

Class location: Digitally on Zoom, at least for now. The Zoom meeting ID is 922 8835 4049, or click here: https://NTNU.zoom.us/j/92288354049. You will need a password also; contact Peder directly for this.

Class times: 10:15–12 on Monday and Tuesday each week; see the detailed schedule below.

Reference group

  • Members: Markus Hagen, Therese Strand, Torgeir Aambø
  • First meeting: 5 February
  • Second meeting: 24 March
  • Third meeting: 7 May
  • You can read a summary of the meetings here: download summary. As always, if you have any suggestions, comments, or requests regarding the course, you are welcome to contact me directly, or talk with one of the reference group members so that they can pass along your comments anonymously.

Digital class expectations

Due to the current infection regulations, class will be held fully digitally on Zoom, at least for the time being. However, my aim is to create a digital classroom that closely reflects the experience of an in-person classroom. In order to do this, the following are some of the expectations.

  • Leave your video on during class, but mute yourself unless you are speaking.
  • Make sure your name appears correctly in your Zoom profile (to help facilitate conversations during class).
  • Expect to interact with me and others during class.
  • Come on time and plan to participate for the full duration of class (and if there is some reason this is not possible, let me know in advance).
  • Avoid other distractions during class (for example: phone, TV, friends, etc.)
  • I will not be recording class meetings, for a number of reasons: much of the benefit of class will be in active participation, class time will not only be lecture, and I want to encourage everyone to participate and feel free to make mistakes without being recorded. If you won't be able to attend class, ask a friend to take notes.
  • In general, treat our digital classes in the same way you would an in person class.

Exam

Details about the final exam:

  • The exam will be held digitally on Zoom.
  • Dates: The exam will take place 11-12 May or 19-21 May.
  • Sign up for a final exam time at this link: https://doodle.com/poll/tnu5dkkasp7webui?utm_source=poll&utm_medium=link Please choose only one time, and enter your full name. The deadline is Wednesday 5 May. If none of the times work for your schedule, please contact me.
  • Format: The exam is an oral exam (the only participants will be me, you, and one other examiner). The exam has two parts:
  1. Part 1: (around 10 minutes) You will prepare and present the solution/proof to one problem/result from the course that you found particularly interesting or enlightening. You may not be able to include all details, however you should be able to explain the details if asked. It is also acceptable to present (part of) a paper/text not from the course but including the same material.
  2. Part 2: (around 30 minutes) You will be asked to solve/prove a number of exercises/results from the course. The number and levels of questions will vary, but they will be drawn from exercises, course notes and text, and material we covered during lecture.
  • Note: You should come prepared to present your problem/result for Part 1 at the start of the exam. However, you should NOT use pre-made notes/slides for this part. Although you should certainly prepare your thoughts in advance and practice your explanation, your presentation should be without notes. Imagine you are just explaining a result on the board for me (not reading from notes), and that I will probably have questions for you along the way.

I have created a document with further details regarding the content of the final exam: Details on final exam content.

Scope / syllabus

We follow the classical approach to algebraic geometry while discussing some more modern trends. We will closely follow Daniel Perrin's book, with tentative goals including: Bézout's Theorem, sheaf cohomology, and the Riemann-Roch Theorem.

Requirements are basic knowledge of (commutative) ring theory and point-set topology.

Book

The course will follow the book Algebraic Geometry - An Introduction by Daniel Perrin. This book is originally written in French. However, I will be using the English translation.

You should be able to get the book at akademika, or a quick google search will probably lead you to a pdf…

I will note that the very motivated student should obtain the canonical textbook by Hartshorne for further study, but it will not be needed for this class.

Plan / log

We will closely follow Daniel Perin's book, all page and section numbers refer there.

Date Topics Material
12.1 (Tuesday) Introduction: objects, problems, and motivation. Discuss meeting times. (download 12 Jan notes) Pages 1-8
tentative schedule
18.1 (Monday) Affine algebraic sets, Zariski topoplogy, Ideal of an affine algebraic set (download 18 Jan notes) I.1,2
19.1 (Tuesday) Irreducibility, Hilbert's nullstellensatz (download 19 Jan notes) I.3,4
25.1 (Monday) Applications of Hilbert's nullstellensatz, first step towards Bezout (download 25 jan notes) I.4,5
26.1 (Tuesday) Regular maps between affine algebraic sets, projective space (download 26 Jan notes) I.6, II.1
1.2 (Monday) Sign up to present a problem from ch. I: click to enter your selection I Exercise day
2.2 (Tuesday) Projective space: what does it look like? (download 2 Feb notes) II.1,2,3
8.2 (Monday) Projective algebraic sets, projective Nullstellensatz, graded ring associated to projective algebraic set (review 7: graded rings on your own) (download 8 Feb notes) II.4,5,6,(7)
9.2 (Tuesday) Introduction to sheaves (download 9 Feb notes) III.0,1
15.2 (Monday) Sheaves - stalks, germs, examples (restriction, constant, pushforward, skyscraper) (download 15 Feb notes) Hartshorne book II.1, Ravi Vakil's notes ch. 2
16.2 (Tuesday) Sheaves - sheaves of rings, ringed spaces, morphisms of sheaves, sheafification, kernel/cokernel/image presheaves (download 16 Feb notes) Hartshorne book II.1, Ravi Vakil's notes ch. 2
22.2 (Monday) Sheafification, Sheaf kernel/image/cokernel, exact sequences of sheaves, Sheaf Hom, overview of affine/projective varieties (download 22 Feb notes) Hartshorne, III.27
23.2 (Tuesday) Structure sheaf on an affine algebraic set, affine algebraic variety, projective algebraic variety (download 23 Feb notes) III.2,3,4,5,6,8
1.3 (Monday) Sign up to present a problem from ch. II: click here to enter your selection II Exercise day
2.3 (Tuesday) Finish chapter III: sheaves of modules on affine and projective algebraic varieties (download 2 March notes) III.7,8,9,10
8.3 (Monday) Dimension of algebraic varieties, relation to Krull dimension (download 8 March notes) IV.1
9.3 (Tuesday) Krull's principal ideal theorem, systems of parameters (download 9 March notes) IV.2
15.3 (Monday) Exercise day - chapter III: present solutions to (1)-(5), B2, and more - record solutions in Overleaf III Exercise day
16.3 (Tuesday) Dimension of morphisms and relation with fibers, dimension theorem (download 16 March notes) IV.3
22.3 (Monday) Tangent spaces: deformations, derivations, and examples (download 22 March notes) V.1
23.3 (Tuesday) Tangent spaces: singular points, Jacobian criterion, regular local rings, curves (download 23 March notes) V.2,3,4
29.3 (Monday) break - no class
30.3 (Tuesday) break - no class
5.4 (Monday) break - no class
6.4 (Tuesday) Independent work day - no class meeting. Use this time to work on chapter IV and V exercises (in the file posted below). You can either work independently, or collaborate with others from class.
12.4 (Monday) Bezout's Theorem: intersection multiplicities (download 12 April notes) VI.1
13.4 (Tuesday) Bezout's Theorem and proof (download 13 April notes) VI.2
19.4 (Monday) Exercise day - chapter IV and V: present solutions IV, V
20.4 (Tuesday) Sheaf cohomology: an overview of Cech cohomology and vanishing theorems (you may want to review homological algebra found VII, section 1 before class - I will assume this background) (download 20 April notes) VII
26.4 (Monday) A bit more about cohomology of twisted sheaves, Euler-Poincare characteristics, degree and genus of projective curves, Riemann-Roch Theorem (download 26 April notes) VII, VIII
27.4 (Tuesday) Schemes - in general (download 27 April notes) Hartshorne book, chapter II.1-5
3.5 (Monday) Schemes - in general (download 3 May notes) Hartshorne book, chapter II.1-5
4.5 (Tuesday) Exercise day - wrap up on all remaining exercises and questions VI, VII, VIII, schemes

Exercises

The exercises are posted here (click to download). Last update: 7 May. I will periodically update the file. Dates to present will be posted on the schedule above. We will discuss solutions to exercises roughly one class period every other week.

As a student, you will present a solution to at least one of the problems during the semester.

Some solutions will be collected here, written by students: https://www.overleaf.com/6584828773msfshpttcbpm (this link should give you access to edit). In short, if you present a solution during class, then go and record your solution in this file.

The final exam will substantially draw from questions on the exercises.

2021-05-11, Peder Thompson