Schemes and divisors

(or how to translate intuition into algebraic geometry)

(or how to translate intuition into algebraic geometry)

Wintersemester 2017, JGU Mainz

The lectures are given on

**Mondays, 14 - 16** in **Room 04-522** and on

**Thursdays, 10 - 12** in **Room 03-424**.

In 1960, Alexander Grothendieck introduced the notion of Scheme in his treatise
*Éléments de géométrie algébrique*.
This notion developed a new formalism that led to the proofs of many long standing problems in algebraic geometry,
such as, among others, the Weil conjectures and Fermat’s Last theorem, starting a new era for algebraic geometry.
It also led to more rigorous proofs of classical results in algebraic geometry, like Bertini’s theorem.

The aim of this course is to make acquaintance with the definition of a divisor of a scheme.
We will see how divisors are important to study maps of a given variety into a projective space,
and also to study families of varieties.
In doing so, we will follow Hartshorne's book *Algebraic Geometry*, the second half of Chapter II.
If time permits, we will also introduce the notion of formal scheme.

We shall follow Hartshorne's *Algebraic Geometry* [HAG], the second chapter.

The interested student may also read *The red book of varieties and schemes*, by Mumford [MVS].

For reference to commutative algebra, we suggest the book *Introduction to Commutative Algebra*, by Atiyah--MacDonald [AM], and/or
*Commutative Algebra with a view toward Algebraic Geometry* by Eisenbud [EIS].

The exam will be an oral exam at the end of the course.

Date | Topics | Lecturer | Homework |
---|---|---|---|

Th. October 19 | Weil divisors | Davide | Exercises |

Mo. October 23 | Discussion on the exercises | Davide | |

Th. October 26 | Examples of Weil divisors and definition of Cartier divisors. | Davide | Exercises |

Mo. October 30 | Discussion on the exercises | Davide | |

Mo. November 6 | Cartier divisors and invertible sheaves | Davide | Exercises |

Th. November 9 | Discussion on the exercises | Davide | |

Mo. November 13 | Serre's twisting sheaves | Davide | Exercises |

Th. November 16 | Very ample divisors | Davide | |

Mo. November 20 | Ample divisors | Davide | |

Th. November 23 | Divisors on curves | Davide | |

Mo. November 27 | Local systems | Davide | Exercises |

Th. November 30 | Introduction to the study of curves | Davide | |

Mo. December 4 | Definition of Proj |
Dino | Exercises |

Th. December 7 | Associated projective space bundle | Dino | |

Mo. December 11 | Blow up | Dino | Exercises |

Th. December 14 | Properties of the blow up | Dino | |

Th. December 21 | Discussion on the exercises | Dino | |

Mo. January 8 | Kähler differentials | Dino | |

Th. January 11 | Sheaves of differentials | Dino | Exercises |

Mo. January 15 | Nonsingular varieties | Dino | |

Th. January 18 | The geometric genus | Dino | Exercises |

Mo. January 22 | Kodaira dimension | Dino | |

Th. January 25 | Discussion on the exercises | Dino | |

Mo. January 29 | Discussion on the exercises | Dino |