This semester we will work through a large part of FGA Explained. This book is written to expand on and clarify the material of Grothendieck's influential FGA papers (wiki, links to the original articles), which are rather terse.

The main topics we will cover are:

- Representability of Hilbert and Quot functors;
- Local study of Hilbert and Quot: (elementary) deformation theory;
- Representability of the Picard functor.

We will skip the part on stacks and fibred categories, hoping that this will not be necessary for reading the rest of the book. We will also skip chapters 7 and 8, which both feel like a bit of a departure from the rest of the material. For a proof of Schlessinger's theorem, we refer to Artin's book Lectures on Deformations of Singularities

Another source that contains chapters on many of the things that we study is Néron Models by Bosch, Lütkebohmert, and Raynaud. For example, it has a great chapter on descent theory, and also one on the Picard functor.

**Prerequisites:** A basic command of algebraic geometry is assumed, along the lines of Hartshorne's book. Moreover, we will freely use Grothendieck topologies and sites; notably the étale and fpqc site. In particular, fpqc descent is often needed. Following Prof. de Jong, we will assume that all commutative algebra is trivial.

The language of category theory is essential. Participants are assumed to be familiar with the notions of functors, natural transformations, etc. Since one of the recurring themes this semester is representability of functors, the Yoneda lemma is essential from a very early stage.

**Homework:** Participants are strongly encouraged to do weekly reading, to keep up with the vast amount of theory we have to cover. The book also contains some exercises, which you may find useful.

**Organiser:** Remy van Dobben de Bruyn.

**Time and location:** Tuesday 16:30-18:00 in Math 507.

Date | Speaker | Title | Ch. |
---|---|---|---|

19 Jan | R. van Dobben de Bruyn | Introduction: representable functors. Outline of topics. | |

26 Jan | R. van Dobben de Bruyn | The Hilbert and Quot functors. Examples. | 5.1 |

2 Feb | R. van Dobben de Bruyn | Outline of construction; Castelnuovo–Mumford regularity | (5.5), 5.2 |

9 Feb | Shizhang Li | Semi-continuity and base change | 5.3 |

16 Feb | Dmitrii Pirozhkov | Flattening stratification | 5.4 |

23 Feb | Qixiao Ma | End of the proof. Applications. | 5.5, 5.6 |

1 Mar | Carl Lian | Infinitesimal study of schemes | 6.1 |

8 Mar | Dan Gulotta | Pro-representable functors | 6.2, 6.3 |

15 Mar | Spring break |
||

22 Mar | Shizhang Li | Schlessinger's theorem | Artin 1.7 |

29 Mar | Monica Marinescu | Examples of tangent-obstruction theories | 6.4, 6.5 |

5 Apr | Yogesh More (SUNY) | The several Picard functors | 9.2 |

12 Apr | Dmitrii Pirozhkov | Relative effective divisors | 9.3 |

19 Apr | Qixiao Ma | Main existence theorem | 9.4 |

26 Apr | Darren Gooden | Variants: Pic⁰ and Pic^τ | 9.5, 9.6 |