# Algebraic Geometry for Number Theory (TCC)

**Organiser: **Florian Bouyer

**Note**: This is a study group and not a TCC lecture course. For a list of TCC course in Autumn click here

This study group will be trying to understand various topics in Algebraic Geometry and how they apply in a Number Theory context, with hopefully some concrete examples. All are welcomed and no pre-requisite in Algebraic Geometry are required (although having a general knowledge of varieties will help).

We aim to use a mixture of sources, including:

- Algebraic Geometry: Robin Hartshorn

- Algebraic Geometry and Arithmetic Curves: Qing Liu

- Rational Points on Varieties: Bjorn Poonen

We will meet on Thursday between 9 and 11, from the 10th of October to 19th of December. The plan is to have a presentation for an hour and use the second hour as a discussion. What follows is a tentative outline of topics, this is very likely to change depending of the demands/suggestions made by people attending. Please contact me if you want to be on the mailing list organising the topics to cover.

**Location:** We meet in the TCC room. In Warwick this is room B0.06.

**Other Resources: **A list of resources that you might also find useful for further reading. Note that they vary in level of difficulty (from easy to "comprehensive").:

- Goerge R. Kempf:
*Algebraic Varieties* - Ravi's notes: http://math.stanford.edu/~vakil/216blog/ (Find the most recent version)

- http://stacks.math.columbia.edu/

### Schedule:

Date |
Speaker |
Topic |
More Details |

10th October |
Florian Bouyer |
Introduction and Sheaves. |
Qing Liu pg 26-29,33-38. Some examples from Hartshorne |

17th October | Chris Williams | Schemes and Algebraic Varieties |
Qing Liu pg 42-44. My typed up notes. Slides for the week |

24th October | Pedro Lemos |
More on Schemes |
Hartshorne. My typed up notes. Slides for the week |

31st October | Basic Properties of Schemes | (e.g. Integral, Neotherian, etc) | |

7th November |
Adam Morgan |
Sheaves of Modules (Part 1) |
Hartshorne, Chapter II sect. 5. |

14th November | Adam Morgan | Sheaves of Modules (Part 2) | Hartshorne, Chapter II sect. 5 My typed up notes. Slides for the week |

21st November | Celine Maistret |
Divisors | Hartshorne, Silverman. My typed up notes. Slides for the week |

28th November | Ian Vincent | Morphisms and Blowing Up | My typed up notes. Slides for the week |

5th December |
Samuele Anni | Minimal Models and Integral Models |
My typed up notes. Slides for the week. Note that algorithm to find minimal models have been implemented this september, for genus less than and equal to 2. |

12th December | Jeroen Sijsling |
More minimal models |
My typed up notes. Slides for the week |

19th December | Jeroen Sijsling | More minimal models | My typed up notes. Slides for the week |