# 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
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
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.