Prof. Dr. U. Görtz, Dr. B. Banwait

## Seminar on Algebraic Surfaces

We will study algebraic surfaces, more precisely smooth projective varieties of dimension $2$ over an algebraically closed field. While the restriction to dimension $2$ simplifies many things in comparison to the general theory, the situation is already considerably more complicated than for algebraic curves, i.e., $1$-dimensional varieties. Therefore, *algebraic surfaces* are a natural topic to study. On the one hand, it is exciting and interesting on its own, with a number of geometrically appealing results, on the other hand the case of surfaces can serve as a helpful guide line for many questions about higher-dimensional varieties.

We will focus particularly on understanding some interesting and important classes of examples of algebraic surfaces.

In the course of the seminar, we will see how to make use of the modern machinery of schemes and cohomology to tackle, among others, several beautiful classical results in algebraic geometry.

Mostly we will follow Chapter V of Hartshorne’s *Algebraic Geometry*. See the program for further references.

**Seminar Program**: pdf

**Prerequisites:** Good knowledge of algebraic geometry, e.g., as covered in J. Heinloth’s courses Algebraic Geometry I and Algebraic Geometry II. [Hartshorne] Ch. II + Ch. III + Ch. IV.1 is definitively more than enough.

**Organizational remarks:** It is expected that you give a blackboard talk. You should plan for a talk duration of 80 minutes or less. Afterwards all participants will have the occasion to ask questions and/or give feedback. Please take preparation of your talk very seriously, and start early on! B. Banwait will be available to help you with questions. You should see him at least once, well in advance of your talk. An **active** participation in the seminar is expected.

For a successful talk, you will obtain 9 ECTS points (Master Seminar).

Upon your choice, you can give the talk in German or in English.

**Contact:** ulrich.goertz@uni-due.de

## Talks

Title of talk | Speaker | |

1 | Divisors and line bundles, ampleness | U. Görtz |

2 | Serre duality | B. Banwait |

3 | Zariski’s main theorem | B. Banwait |

4 | Cohomology and base change | U. Görtz |

5 | The intersection pairing on an algebraic surface | Shen-Ning Tung |

6 | The theorem of Riemann-Roch for surfaces | Felix Gora |

7 | The Hodge index theorem and the Nakai-Moishezon criterion | Felix Gora |

8 | Monoidal transformations | Yisheng Tian |

9 | Cubic surfaces I | Yisheng Tian |

10 | 11.2.: Cubic surfaces II: The 27 lines | Barinder Banwait |