# MA8203 Algebraic Geometry

## Lecturer

**Steffen Oppermann**, room 844, Sentralbygg II, Steffen [dot] Oppermann [at] math [dot] ntnu [dot] no

## Schedule

Lectures will regularly take place **Mondays**, **14:15–16:00** in room **656**, and **Tuesdays**, **10:15–12:00** in room **822**.

Exceptions | |
---|---|

Tu, Jan 18th | lecture at 8:15 in 656 |

Mo, Feb 21st | no lecture |

Tu, Feb 22nd | no lecture |

Th, Mar 3rd | extra lecture at 12:15 in 656 |

Th, Mar 10th | extra lecture at 12:15 in 656 |

## Book

The course will follow the book *Algebraic Geometry - An Introduction* by **Daniel Perrin**. (We will not cover the entire content of this book.)

This book is originally written in French. However I will be using the English translation.

Individual chapters of the book can be downloaded from SpringerLink via the NTNU library (i.e. you have to be on campus or surf via campus).

## Exercises

Date | Problems |
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Jan 11 – Jan 17 | Consider the following subsets of \(\mathbb{R}^2\): \(A = \{ (t, \text{sin } t) \mid t \in \mathbb{R} \} \) and \(B = \{ (t, \frac{1}{t}) \mid t \in \mathbb{R} \setminus \{ 0 \} \} \). Are these sets affine algebraic? If yes: Are they irreducible? If no: what is the Zariski closure? Exercises 2, 3, and 7 from Chapter I in the book Read the introduction to the book |

Jan 25 – Jan 31 | Exercises 1 and 2 from Chapter II in the book Read Chapter II.2 |

Feb 01 – Feb 07 | Exercise 4 from Chapter II in the book Take another look at sheafification and try to understand what is going on. |

Feb 08 – Feb 14 | Exercises A-2 and A-3 from Chapter III in the book. |

Feb 15 – Feb 28 | Exercises A-4 and A-8 from Chapter III in the book. |

Mar 03 – Mar 07 | Exercises A-7 and B-5 from Chapter III in the book. |

Mar 10 – Mar 14 | Exercise B-3 |

Mar 15 – Mar 21 | Read page 77 in the book, that is look at examples motivating a relationship between the dimension of fibers of a morphisms to the dimensions of the two varieties |

Mar 22 – Mar 28 | Exercises 3 and 4 from Chapter IV in the book. |

Mar 29 – Apr 04 | Let \(f \in k[X_1, \ldots, X_n]\) of degree 3. a) Show that if \(V(f) \subseteq k^n \) contains two singular points, then it also contains the line through these two points. b) Assume \(n = 2 \), and that \(V(f)\) contains at least three singular points not lying on a common line. Show that \(V(f)\) is the union of three lines. c) Give an example showing that a) does not hold for degree f = 4. Exercise 8 from Chapter V in the book. (Hint: you may use Corollary IV 2.11(2).) |