Algebraic geometry II - schemes and sheaves

Generalities:

The lectures and the exercise sessions will meet as follows:

Wednesdays in MA A3 30: lecture 12:00-14:00, exercise session (Questions and answers session) 14:00-16:00;

Fridays in GR A3 31: lecture 13:00-15:00, exercise session (Presentation session) 15:00-17:00.

Final exam: 

January 13th, 2026. Between 9am-12 noon. Venue: CM1106.

Homework: 

There will be two types of homeworks:

1. Exercise to present. Together with the TA (to avoid overlaps with other students), you choose one of the exercises from the current exercise sheet that is not an exercise to be handed in. Then you agree with the TA at which exercise session you would present it (this will almost always be the subsequent week). And finally you present the exercise at the board during the agreed exercise session. You will be required to present 2-4 exercises during the semester, where the final number will be decided once attendance is stabilized. If you present the exercise on a reasonable level, then you get maximum points. If not, then you get a chance to redo it. If you do not redo it, you get a 0. For the first week on Friday, the TA will take care of presenting exercices. From week 2, students will present exercices as explained above. The TA may also present some exercices on subsequent weeks.
   
2. Exercise to hand in. You will be required to hand in 6 exercises during the semester. This seems to be not so much, but you should prepare yourself for a bit of work, as exercises tend to be sometimes demanding. These hand in exercises will be from the exercise sheets. The solutions should be in pdf, compiled from LaTeX, and they should be handed in here on Moodle. If the homework is not done in LaTeX, then it is worth 0 pts. The deadlines are taken very seriously. That is, the system does not accept the homework even a second later than the deadline.  You are welcome to discuss about the solutions of the exercises, including the ones to be handed in, among each other and also with the TA. However, you have to write them out separately, meaning that the effort of the mathematical redaction should be strictly individual. Copied homeworks are given 0 pts for both sides. 

Final grade: 

The overall score in the class, which will determine the final grade, will be computed as follows: 30% will be from the combination of the two types of homework (5% from the presentations, and 25% from the handed-in exercises), and 70% will be from the written final exam. 

Sources: 

We suggest you use these sources to complement the lecture and the exercises, if needed.

• The main reference for this course is Algebraic Geometry by Hartshorne. 
• Here are Zsolt's notes from when he taught the course. These add details and provide corrections to some parts of Hartshorne. 
• Ravi Vakil's book "The rising sea"(it is an extremely good source to build up the intuition that is not dealt with in Hartshorne, besides it is very thorough and precise) 
  
• Mumford: The red book of varieties and schemes (you need to be on vpn or on campus or the link; this also has more intuition than Hartshorne by not sacrificing preciseness, but it covers certainly way less)
• Eisenbud-Harris: The geometry of schemes (you need to be on vpn or on campus or the link; this is a book primarily aimed to build intuition, covering similar amount of material as that of Mumford's but not in the traditional style of a definition/statement/proof book, but rather it is as someone was explaining scheme theory as a bed time story)
• Görtz-Wedhorn: Algebraic geometry I (you need to be on vpn or on campus or the link; this one has more details than Hartshorne, not that much more intuition)
• Görtz-Wedhorn: Algebraic Geometry II (cohomology). Follow-up to the first book, with detailled proofs on a lot of topics in cohomology of schemes. 
• The recent Algebraic Geometry: Notes on a course by the eminent Michael Artin. Very pedagocical. Works only over complex numbers and does not talk about schemes but contains very good chapters on Cohomology and Riemman-Roch.
  
• Qing Liu: Algebraic Geometry and Arithmetic Curves (all the content is included in the other ones, but it has a nice arithmetic point of view)
• Stacks project: An open source textbook and reference work on algebraic geometry. It is a precise reference on everything in basic (and far more) algebraic geometry. Also, it contains a very detailed exposition of the commutative algebra needed for algebraic geometry.
  
• Mumford-Oda: Algebraic Geometry II (a penultimate draft is available under following the link).
• Exercises of Huybrechts: Exercises for a similar course in Algebraic Geometry given by Daniel Huybrechts.

Discussion forum:

There is a Ed Discussion associated to the course, see the link below. You can use it for questions and discussions about the material of any part of the course, including the lectures and the exercise sessions. Please do not use the Ed Discussion for other purposes.

Advice:  

It is important that you follow the course and the exercise session constantly, that is, you re-read the material of the previous class between two classes, and you solve also those exercises that are not required to be handed in.

Prerequisites

The courses Rings and Modules and Algebraic Geometry I : Curves are prerequisites to this class. You can access the course material for these classes from the academic year 2024-2025 via EPFL's Moodle archive or below.

Hi guys, this week we shall construct fiber products and study certain global properties of schemes (irreducible, reduced, normal, etc) and of morphisms (finite-type, finite, affine, etc)

In the first lecture this week, I will discuss key properties of flat morphisms—such as generic flatness and the openness of flat maps—and explain the criterion for when Proj(S) is flat over 
Spec(S_0). I will also introduce the idea of generic properties of morphisms, showing that certain properties which hold over the generic point must hold on some open subset.

In Friday’s lecture, I will describe the role of discrete valuation rings (DVRs) in algebraic geometry and use them to define and motivate the notions of separated and proper morphisms.