Abelian Varieties, Autumn 2014

WONDER course

B.J.J. Moonen, Radboud University Nijmegen. Email: B.Moonen at -science.ru.nl-

The lectures are given on Thursdays, 2pm till 5pm. The course starts in week 38 (September 18) and continues until week 51 (December 18). In week 37 there is a short mastermath course on categories. If you're not yet familiar with the notion of a category, you're strongly advice to follow this short course.

The lectures are at campus De Uithof in Utrecht. NOTE: The lecture rooms have changed: On October 30 we are in the Minnaert building, room 012; from November 6 on we are in the Minnaert building, room 027.

The course is intended as an advanced course in Algebraic Geometry. We assume familiarity with basic Algebraic Geometry, to the level where you can read Chapters 2 and 3 of Hartshorne's book. We shall use the language of schemes, though in the beginning of the course it suffices if you're familiar with the more classical notion of an algebraic variety. It is highly recommended to follow, in parallel with this course, the mastermath course 'Advanced Algebraic Geometry' taught by Robin de Jong and Lenny Taelman.

We shall not follow one specific text. Most of what I shall discuss can be found in the book-to-be 'Abelian Varieties' that I'm writing jointly with Bas Edixhoven and Gerard van der Geer. Here you can find the first 12 chapters (which is far more than we shall be able to cover). Further there is Mumford's book on Abelian Varieties; note, however, that the way we shall treat duality (one of the central topics of the course) differs from Mumford's approach.

Date | Topics |
---|---|

Sept 18 | Vector bundles, line bundles, divisors, HAG Corollary II.6.16. Read HAG, Chap II, Sections 5 and 6. (For now, we shall only need to work with smooth proper varieties.) Exercises |

Sept 25 | Group varieties. Definition of an Abelian variety. Freeness of the tangent bundle. Rigidity lemma and applications. |

Oct 2 | Elliptic curves. The functor of points of a scheme; group schemes seen from this perspective. Exercises |

Oct 9 | Rigidified line bundles. The Picard scheme. First applications to abelian varieties: the theorems of the cube and of the square. |

Oct 16 | Projectivity of abelian varieties. |

Oct 23 | Lecture by Johan Commelin: basic results about group schemes. |

Oct 30 | Quotients by finite group schemes. Isogenies. |

Nov 6 | More on group schemes. Applications to isogenies. Frobenius isogeny. The p-rank. |

Nov 13 | The homework assignment is now available. Due date: Nov 27. Today we have made a start with the duality theory of abelian varieties; see for instance chapters 6-7-... of the book project. |

Nov 20 | No lecture! |

Nov 27 | Two duality theorems. Hom-schemes. Definition of a polarization. |

Dec 4 | Poincaré splitting theorem. Complex abelian varieties. |

Dec 11 | Jacobians of curves (start). Symmetric powers of curves. |

Dec 18 | Jacobians of curves (continued). |

On January 29 there will be final oral exams in room 207 of the Minnaert building (campus de Uithof). You can only do an oral exam if you have done the midterm assignment. The available slots in the following table are assigned on a first-come, first-go basis; please contact me by email to make an appointment.

Time | Student |
---|---|

10:30-11:00 | |

11:00-11:30 | Emma Brakkee |

11:30-12:00 | Wibrich Drost |

12:00-12:30 | Ties Laarakker |

13:00-13:30 | Stefano Nicotra |

13:30-14:00 | Wessel Bindt |

14:00-14:30 | Jeroen Hanselman |

14:30-15:00 | Tom Wennink |

15:00-15:30 | Ane Anema |

15:30-16:00 |

To the webpage of Ben Moonen.