Lecturer

Johannes Schmitt

Coordinator

Tim Gehrunger

News

• May 28, 2026: Here is a list of example questions for the oral exam. They are representative of the kind of questions I will ask, and the first question of the exam will be from this list. If any of these example questions are unclear or you think there is some problem, please tell me soon - I am happy to clarify or address any such issues. This version incorporates the previous corrections to Exercises 4 and 6.
• Feb 11, 2026: The course website for FS 2026 is online. Schedule and exercise information will be updated continuously during the semester.

Content

The lecture follows very closely the lecture notes by Andreas Gathmann. In particular it covers

• Affine varieties over algebraically closed fields
• Abstract varieties as ringed spaces
• Projective varieties, Grassmannians
• Birational maps and the 27 lines on the cubic surface
• Schemes
• Quasi-coherent sheaves
• Differentials
• Cohomology of sheaves

Prerequisites

Properties of commutative rings and modules over them (as covered by the lecture Commutative Algebra).

Format

The lecture is offered in a flipped classroom setup:

• Lecture videos (weekly)
  • YouTube playlist
  • ETH video portal (ad-free)
  • cover the material of the lecture (whiteboard notes linked below)
  • published Friday, should be watched until the end of the following week
  Goal:
  • present content of the lecture in easily digestible chunks (videos typically 10 - 20 min)
  • students can watch at their own pace (with pauses vs. watch at x 1.5 speed vs. simply reading the notes)
• Live problem-solving sessions (Monday, 14:15 - 15:00, HG G 5)
  • students work on short, accessible exercises (concerning the material from the previous week)
  • working in groups is encouraged
  • lecturer is around to answer questions or give hints
  • ideally, solutions are presented by groups or individual students at the end of the session
  Goal:
  • getting hands-on experience with the concepts from the lecture
  • working in teams (for fun!)
  • practice for oral exam (the opposite of fun!)
• Informal Q&A sessions (Monday, 15:00 - 16:00, HG G 5; optional)
  • optional time after problem sessions where the lecturer stays around for questions, etc.
  Goal:
  • asking questions on the lecture or exercises
  • watching some of the lecture videos
  • talking to other students or working on exercise sheets
• Exercise class (Tuesday, 15:15 - 16:00, HG D 7.1)
  • exercise sheets contain more involved tasks that might need a bit of thinking
  • cover new examples and properties of the concepts from the lecture
  • solutions can be handed in for feedback
  • We strongly encourage you to work on the exercise sheets!
  Goal:
  • gain further practice with lecture material by individually working on exercises
  • getting feedback on arguments and level of understanding

Feedback

Since the format of the course is pretty new, I am always happy for feedback! You can submit it either via email or via this (anonymous) form.

Lectures

Overview

You find the lecture videos on the YouTube playlist or ad-free on video.ethz.ch.

Whiteboard notes

• 01 Affine varieties
• 02 The Zariski topology
• 03 The sheaf of regular functions
• 04 Morphisms
• 05 Varieties
• 06 Projective varieties I - Topology
• 07 Projective varieties II - Ringed spaces
• 08 Grassmannians
• 09 Birational Maps and Blowing up
• 10 Smooth varieties
• 11 The 27 lines on a smooth cubic surface
• 12 Schemes
• 13 Sheaves of modules
• 14 Quasi-coherent sheaves
• 15 Differentials

Exercises

New exercise sheets and sample solutions will be posted here progressively during the semester. We expect you to look at the problems over the weekend and on Monday and to prepare questions for the exercise class on Tuesday.

Please hand in your solutions by the following Friday at 12:00 in the coordinator's box in HG J68. Your solutions will usually be corrected and returned in the following exercise class or, if not collected, returned to the box in HG J68.

Alternatively, you can send your solution attempts by email to tim.gehrunger@math.ethz.ch.

Exercise classes

Literature

• Algebraic Geometry - lecture notes by Andreas Gathmann (main reference)
• The Rising Sea - Foundations of Algebraic Geometry - lecture notes by Ravi Vakil
• Algebraic Geometry - book by Robin Hartshorne (Springer 1977)