# Math 40510: Introduction to Algebraic Geometry Spring, 2020

Instructor: Juan Migliore
Office: HAYE 236
Phone: 631-7345

Office hours:
Tuesday 2:00-3:00,
Thursday 1:00-2:00,
Or by appointment.

Time and place of class: MWF 11:30-12:20 in HAYE 229.

Textbook: Ideals, varieties and algorithms, 4th Edition by David Cox, John Little and Donal O'Shea, Springer-Verlag.

You've already gotten several emails from me about how we are going to manage the transition with zoom. The biggest adjustment will be to move from writing on the blackboard to writing on my iPad. The good news is that I won't have to erase the blackboard.

Here are the notes from all of the lectures this semester. After every class I'll post that day's lecture for the rest of the semester.

Here are the videos of the lectures after spring break. After every class I'll post that day's lecture for the rest of the semester.

Other than that, things should go through pretty much without any major changes. You'll get the same lectures, have the same problem sets with the same deadlines. We will figure out how to handle the final exam soon.

I hope you stay safe, and I hope that the rest of the semester goes as smoothly as possible for all of us.

Problem Sets (Make sure you have the current version!) The links will be made active over the course of the semester.

Course overview: Algebraic Geometry is the study of systems of polynomial equations and their vanishing loci. This field has important components that lie in the realm of geometry, of algebra and of computation (among others) and countless applications. This course tries to give a flavor of these different aspects of the field and how they fit together. Indeed, much of the fascination of this subject comes from the myriad ways in which arguments squarely in one realm give surprising consequences that fall squarely in a different realm.

Commutative Algebra is a closely related field. This course will also be an introduction to Commutative Algebra, with an emphasis on the interplay between the two fields. We will review the basic notions from algebra (e.g. polynomial rings, ideals, quotient rings), and we will discuss the corresponding geometric objects.

Here are some books that might be useful as supplemental reading. Out of courtesy to your classmates, please do not take these out for more than a few days. By far the most important reference is your textbook, but these offer interesting side reading.

• An invitation to Algebraic Geometry by Smith, Kahanpää, Kekäläinen and Traves, Springer-Verlag.
• Computational algebraic geometry, by Schenck, London Mathematical Society.
• Algebraic geometry: a problem solving approach, by Garrity, Belshoff, Boos, Brown and Lienert, American Mathematical Society.
• Undergraduate Algebraic Geometry, by Reid, London Mathematical Society.
• Elementary Algebraic Geometry, by Kendig, Springer-Verlag, GraduateTexts in Mathematics.<\li>
• Rudiments of Algebraic Geometry, by Jenner, Oxford University Texts in the Mathematical Sciences.
• Undergraduate Commutative Algebra, by Reid, Cambridge University Texts.
• Algebraic Curves, by Walker, Springer-Verlag.
• Algebraic Geometry: A First Course, by Harris, Springer-Verlag, Graduate Texts in Mathematics.
• Elementary Algebraic Geometry, by Hulek, American Mathematical Society.

Examinations, homework and grades. There will be three problem sets and a final exam. The problem sets can be accessed above. They will be updated regularly. I will let you know when the current version is in final form. Do not assume that the version you see now is in final form before I say that it is!!!

• Problem Set 1: due Friday, Feb. 14, 2020, in class.
• Problem Set 2: due Friday, March 20, 2020, in class.
• Problem Set 3: due Friday, April 24, 2020, in class.
• Final Exam: Monday, May 4, 4:15-6:15 PM, location TBA. (Take this into account when you make your travel plans!)
How you will be evaluated: Your course grade will be based on your total score out of 450, with points allocated as follows:
• Problem sets: Each one will be worth 100 points of your final grade.
• Final exam: Worth 150 points.

Homework and Reading: There will not be any homework assigned apart from the problem sets. In my lectures I will try to follow the material in the textbooks, although not all of it and not necessarily in the order in which it is presented there. There may also be some additional material not in the books. The problem sets will be based primarily on class lectures, but possibly also from the material in the chapters. Thus it is important to read the material as well as following the lectures.

Honor Code: Both the final exam and the problem sets are conducted under the Notre Dame Honor Code. On the first day of class I will give you a handout and I will give you very clear explanations of what is expected of you in the course and how the Honor Code applies to your work. This is important information, and if for some reason you are not in class on the first day, be sure to talk to me about it!! As always, you are strongly encouraged to come talk to me at any time if you are having any problems with anything.