University of Wisconsin-Madison
University of Wisconsin-Madison
Mathematics Department

Math 763
Introduction to Algebraic Geometry
A first course
Lecturer: Andrei Caldararu

Fall 2020

About the course: This is an introduction to the basic ideas and methods of algebraic geometry. It will introduce the main objects of study of the subject, affine and projective varieties, and then we will concentrate on curves, divisors on curves, etc. A secret goal will be to get to state and prove Riemann-Roch for curves. We will try to emphasize examples over the theory.

We will use a fairly algebraic approach, hence a good control of commutative ring theory (at a minimum at the level of 741/742) is needed.

Official syllabus: available here.

Approximate syllabus:

Lectures: There will be two 75 minute lectures per week:

TR 9:30-10:45 online, synchronously, via Zoom. Recurrent Zoom link here. If prompted for a passcode, it is 691792. The Meeting ID is 94188494774.

Video recordings: I will post here recordings of the lectures. Unfortunately Zoom only preserves these for 30 days.

Lecture of Thursday, 9/10, Passcode: vnFtyA2.

Lecture of Tuesday, 9/15, Passcode: aXtu*#v8

Lecture of Thursday 9/17, Passcode: 40M0=Epe

Lecture of Tuesday 9/22, Passcode: $w%G37%3

Lecture of Thursday 9/24, Passcode: &@=syIE8

Lecture of Tuesday 9/29, Passcode: @E8e5tZr

Lecture of Thursday 10/1, Passcode: @%ya5sSa

Lecture of Tuesday 10/6, Passcode: $v4dQsvA

Lecture of Thursday 10/8, Passcode: =ghEHQB4

Lecture of Tuesday 10/13, Passcode: E$2=bnM7

Lecture of Thursday 10/15, Passcode: g42y&d=M

Lecture of Tuesday 10/20, Passcode: 1i*TW$?#

Lecture of Thursday 10/22, Passcode: Cy!Vrrf8

Lecture of Tuesday 10/27, Passcode: 28f7rU&&

Lecture of Thursday 10/29, Passcode: n^DA*H!5

Lecture of Tuesday 11/3, Passcode: 68!R.PUe

Lecture of Thursday 11/5, Passcode: B#^+DR83

Lecture of Tuesday 11/10, Passcode: 5U8%xn&9

Lecture of Thursday 11/12, Passcode: =FY=*59m

Lecture of Tuesday 11/17, Passcode: *7u!P1Bh

Lecture of Thgursday 11/19, Passcode: ?g5%*aB=

Lecture of Tuesday 11/24, Passcode: 05d$buhh

Lecture of Tuesday 12/1, Passcode: LCQ4?nT.

Lecture of Thursday 12/3, Passcode: C73$UKgZ

Lecture of Tuesday 12/8, Passcode: LC=yz5YV

Lecture of Thursday 12/10, Passcode: M@BE1T+%

Text: The textbook for the course is

Algebraic Geometry by Robin Hartshorne, GTM 52 (1977), Springer-Verlag

However, I shall draw inspiration from a few other texts, among them Joe Harris' "Algebraic Geometry" and Shafarevich's books.

You can also consult Milne's notes Milne's notes http://www.jmilne.org/math/CourseNotes/ag.html

Office hours: Tuesdays 1:30-2:30 and by appointment, via zoom. The meeting ID is 98259438606 and the passcode is 653978.

Grading: The term grade will be primarily based on homework. I may decide to assign some final reading and exposition projects. This will be discussed later in the semester.

Homework: Homework will be due a week after it is assigned, via email. All claims that you make in your homework MUST BE PROVED in order to receive credit.

Homework assignments:

Homework assignment 1, due September 25: Hartshorne §1.1: 1, 2, 5, 8, 9, 11

Homework assignment 2, due October 2: Hartshorne §1.2: 4, 6, 11, 13, 14, 15, 16

Homework assignment 3, due October 16: Hartshorne §1.3: 1, 2, 5, 6, 7, 15, 17, 21

Homework assignment 4, due October 23: Hartshorne §1.4: 1, 3, 5, 6, 7

Homework assignment 5, due November 20: Hartshorne §1.5: 1, 3, 4, 7, 9, 10

Also do the exercise assigned in class: blow up the singular point on z^2 - yx^2 +4y^{n+1} = 0 and further singular points on the resulting blown-up surfaces until the resulting surface is no longer singular, and draw the dual graph of the exceptional locus.

Homework assignment 6, due December 10: Hartshorne §1.6: 1, 2, 3, 4, 6; §1.7: 2, 5, 8