Introduction to Algebraic Curves, Autumn 2015

Master course, Radboud University Nijmegen

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

The lectures are given on Thursdays from 2.30 pm (sharp!) till 4.15 pm. The course starts in week 37 (September 10) and continues until the second week of January. There are no lectures in weeks 44-46. **(Note the unusual time.)**

The lectures are in the Huygens building. In weeks 37-42 we're in room HG00.068, in week 43 in HG00.108, from week 47 on we're in HG01.028.

The topic of the course is the geometry of algebraic curves. The course can serve as a first introduction to Algebraic Geometry. We assume familiarity with basic algebra, abstract topology, some complex analysis, and we assume the students have seen the notion of a smooth manifold. In the course we shall use the language of complex manifolds; the basic definitions of this will be discussed.

We shall mainly follow the book 'Introduction to Algebraic Curves' by Ph. Griffiths. (AMS Translations of Mathematical Monographs, Volume 76.) In case you are unable to get it from your library, it is quite easy to get a copy on the internet. If this causes problems, contact me.

Complementary reading: Griffiths-Harris 'Principles of Algebraic Geometry', Chapter 0; Huybrechts's book 'Complex Geometry, An Introduction', especially the first two chapters.

Exercises will be assigned on a bi-weekly basis. These are to be handed in to Johan Commelin (j.commelin at math.ru.nl), either in his pigeon hole (Huygens building, opposite to room HG03.708), or electronically. Handing in by email is possible only if you write your solutions using (La)TeX; in that case, send the pdf output. You are allowed to collaborate with other students but what you write and hand in should be your own work. If different students hand in the same work, we will not accept their work.

As this is a master course, you are supposed to study the details of what is covered in the oral lectures in far greater depth than what is needed to do the hand-in exercises. As we accumulate material, I will assume that you have fully digested what was treated before.

The course is concluded by an exam. Depending on the number of students this will be a written or an oral exam. You can only do the exam if your average grade for the hand-in exercises is 6 or higher.

Date | Topics | Hand-in exercises |
---|---|---|

Sept 10 | Complex function theory, complex manifolds | |

Sept 17 | Tangent spaces, inverse and implicit function thm | Exercise sheet 1 |

Sept 24 | Complex submanifolds. 1-Dimensional complex tori. | |

Oct 1 | Riemann surfaces: Meromorphic functions. Ramification index and degree of a map. | Exercise sheet 2 |

Oct 8 | Structure of the local ring. Plane curves and examples. | |

Oct 15 | Normalization of plane curves. | Exercise sheet 3 |

Oct 22 | Holomorphic and moremorphic 1-forms. Statement of RR. | |

Oct 29 | No lecture | |

Nov 5 | No lecture | |

Nov 12 | No lecture | |

Nov 19 | Riemann-Hurwitz. Intersection of plane curves. | Exercise sheet 4 |

Nov 26 | Intersection numbers of plane curves. Bézout's theorem. The genus of a plane curve. Linear systems of divisors (beginning). | |

Dec 3 | Linear systems and morphisms to projective spaces. The canonical embedding. | Exercise sheet 5 |

Dec 10 | ||

Dec 17 | Weierstrass points and automorphisms | Final exercise sheet |

Jan 7 | ||

Jan 14 |

To the webpage of Ben Moonen.