- Lecturer
- Johannes Schmitt
- Coordinator
- Dmitrii Krekov

- Jul 10: 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.

Edit (Jul 16): In Exercise 4, the ambient space of \(Y\) was changed from \(\mathbb{A}_{\mathbb{Z}}^2\) to \(\mathbb{A}_{\mathbb{Z}}^1\).

Edit (Aug 1): In Exercise 6 b) and c) the "map" f should actually be a "morphism" - this has been corrected, but if you see this edit too late, you can also answer the original question interpreting f as a rational map - Mar 19: I discuss the
**results of the class survey**on the D-MATH forum, and explain the changes in the lecture videos and live problem sessions going forward. - Feb 23: Thanks to everyone who participated in the poll about the best format for a QA-platform for the course! Following the most popular option, there is now a subforum for the course on the D-MATH forum, where you can ask questions on the course material, exercise sheets and organizational issues.
- Feb 16: Both the live problem-sessions and the exercise class
**start in the second week of the semester**, so there is**no class in the first week**! To get an overview of the content and format of the course, you can watch the introduction video (including some small demonstration of the kind of live-problems we'll have in the Tuesday sessions).

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

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

The lecture is offered in a flipped classroom setup:

**Lecture videos (weekly, on Youtube)**- 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)

- cover the material of the lecture (
**Live problem-solving sessions (Tuesday, 14:15 - 15:00, HG D 7.1)**- students work on short, accessible exercises (concerning the material from the previous week)
- working in groups in 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 (Tuesday, 15:00 - 16:00, HG D 7.1; optional)**- optional time after problem sessions where 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 (Monday, 16:15 - 17:00, HG E 1.2)**- 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

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.

You find the lecture videos on youtube.

Week | Material |
---|---|

1 | 01.01 Affine varieties - 02.03 Irreducible affine varieties |

2 | 02.04 Noetherian spaces and irreducible decompositions - 02.09 Dimension theory of reducible spaces |

3 | 03.01 Definition of regular functions - 04.02 Properties of morphisms of ringed spaces |

4 | 04.03 Morphisms between affine varieties - 05.05 Products of prevarieties |

5 | 05.06 Separatedness and the definition of varieties - 06.11 The projective closure |

6 | 06.12 Projective hypersurfaces - 07.06 Complete varieties |

7 | 07.07 The Veronese embedding - 09.01 Rational and birational maps |

8 | 09.02 Rational functions - 09.08 Blowing up to remove indeterminacies of rational maps |

9 | 10.01 The tangent space to a variety - 11.01 The lines on the Fermat cubic surface |

10 | 11.02 The moduli space of smooth cubic surfaces - 12.05 Regular functions and the structure sheaf of affine schemes |

11 | 12.06 Regular functions on distinguished open sets - 12.14 Definition of schemes |

12 | 12.15 Schemes from prevarieties - 13.05 The tensor presheaf |

13 | 13.06 Sheafification - 14.05 Properties of pullback sheaves |

14 | 14.06 Locally free sheaves - 15.04 Application - the genus of a smooth projective curve (optional: Epilogue - What next) |

- 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

The new exercises will be posted here on Fridays. We expect you to look at the problems over the weekend and to prepare questions for the exercise class on Monday.

Please hand in your solutions by the following Friday at 12:00 in your assistant'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.

exercise sheet | due by | solutions |
---|---|---|

Exercise sheet 1 | February 23 | Solution 1 |

Exercise sheet 2 | March 1 | Solution 2 |

Exercise sheet 3 | March 8 | Solution 3 |

Exercise sheet 4 | March 15 | Solution 4 |

Exercise sheet 5 | March 22 | Solution 5 |

Exercise sheet 6 | March 29 | Solution 6 |

Exercise sheet 7 | April 12 | Solution 7 |

Exercise sheet 8 | April 19 | Solution 8 |

Exercise sheet 9 | April 26 | Solution 9 |

Exercise sheet 10 | May 3 | Solution 10 |

Exercise sheet 11 | May 10 | Solution 11 |

Exercise sheet 12 | May 17 | Solution 12 |

Exercise sheet 13 | May 24 | Solution 13 |

Exercise sheet 14 | May 31 | Solution 14 |

live problem sheet | date | solutions |
---|---|---|

Sheet 1 | February 27 | Solution 1 |

Sheet 2 | March 5 | Solution 2 |

Sheet 3 | March 12 | Solution 3 |

Sheet 4 | March 19 | Solution 4 |

Sheet 5 | March 26 | Solution 5 |

Sheet 6 | April 9 | Solution 6 |

Sheet 7 | April 16 | Solution 7 |

Sheet 8 | April 23 | Solution 8 |

Sheet 9 | April 30 | Solution 9 |

Sheet 10 | May 7 | Solution 10 |

Sheet 11 | May 14 | Solution 11 |

Sheet 12 | May 21 | Solution 12 |

Sheet 13 | May 28 | Solution 13 |

time | room | assistant | language |
---|---|---|---|

Mo 16-17 | HG E 1.2 | Dmitrii Krekov | English |

- 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)