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: 

Monday, January 13 2025, from 9:15 a.m. to 12:15 p.m., room MA A1 10.

Notes: 

Besides the above book, we provide LaTeX'd notes here on the  Moodle page of the course, that contain the parts of the material not covered by Hartshorne and add detail (or provide corrections!) to certain parts of Hartshorne.

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. The deadlines are taken very seriously. That is, the system does not accept the homework even a second later than the deadline. More introductory info on LaTex can be found here. If the homework is not done in LaTeX, then it is worth 0 pts. During the exercise sessions as many questions can be asked about the homework as needed. You can also talk with each other about the solutions of the exercises, including the ones to be handed in. 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.

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.

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.

Additional sources: 

We suggest you use these sources to complement Hartshorne if needed:

• Ravi Vakil's notes (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)
• 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.
  

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 2023-2024 via EPFL's Moodle archive.