Information on this page is given in the language of instruction: English, German, or French.

PhD students

• Yi-Hung Chen (2025-)

Wintersemester 26/27

Birational geometry

This lecture is intended as an introduction to the modern methods of birational geometry pioneered by Japanese mathematician Shigefumi Mori in the 1980s. It will consist of four parts:

1. Birational geometry of smooth complex surfaces. (4 lectures)
2. Rational curves — Mori’s Bend-and-Break Method. (4 lectures)
3. Mori’s Cone Theorem. (6 lectures)
4. An introduction to the Minimal Model Program. (1 lecture)

Time and place: Thursdays 8:30 - 10:00 in Room IA1/63.

Prerequisites: Anyone with a background in complex or algebraic geometry is welcome. For curious students coming from algebra and geometry more broadly, Part 1 is self-motivated and can be attended on its own. Some knowledge about the following objects will be assumed throughout the lecture:

• some culture on complex curves/Riemann surfaces;
• (Part 2-4) basic knowledge on schemes;
• (Part 3-4) basic knowledge on vector bundles/sheaves and cohomology.

Commented bibliography:

• Kawamata, Algebraic varieties: Minimal models and finite generation. This is the reference we will follow most closely. The plan is to cover its Sections 1.1 - 1.7, selected topics from 1.13, and Sections 2.1, 2.3, 2.4, 2.7, and 2.8. The book is quite concise, so we will sporadically complement it with the other references below.
• Debarre, Higher dimensional algebraic geometry. This book grew out of lecture notes and offers a more gentle introduction to many of the concepts of the lecture. See notably Chapters 1, 2, 3 on rational curves.
• Kollár—Mori, Birational geometry of Algebraic Varieties. This is the classical reference for the topic. Some of the results presented in this book have since been proven, yet the clear and well-motivated exposition keep it highly relevant. The first chapter gives a particularly nice overview on the Cone Theorem.

Archives

Ruhr-Universität Bochum

• Sommersemester 26: Seminar Topics in Algebraic Geometry (2 SWS, co-organized with Alex Ivanov). Here is the program.
• Sommersemester 26: Algebraische Geometrie II (2 SWS + 2 SWS exercises instructed by Yi-Hung Chen).
• Wintersemester 25/26: Algebraische Geometrie I (4 SWS + 2 SWS Übungen geleitet von Yi-Hung Chen). You can find the exercise sheets here: Week0_ATasteOfTopology ; Week1_AffineAlgebraicSets ; Week2_ProjectiveSpaces ; Week3_ProjectiveAlgebraicSets ; Week4_MapsAndRationalMaps ; Week5_RationalMapsAndSheaves ; Week6_SheafificationSectionsStalks ; Week7_AffineSchemesFiniteSchemes ; Week8_FiniteMorphisms ; Week9and10_BezoutAndDimension ; Week11_DimensionTheory ; Week12_LocallyFreeSheavesGrassmannians ; Week13_SmoothnessSingularities; Week14_RationalityAndEllipticCurves.
• Sommersemester 25: Kommutative Algebra (4 SWS + 2 SWS Übungen geleitet von Emeryck Marie). Hier sind die Übungsblätter verfügbar: Blatt 1 ; Blatt 2 ; Blatt 3 ; Blatt 4 ; Blatt 5 ; Blatt 6 ; Blatt 7 ; Blatt 8 ; Blatt 9 ; Sonderausgabe ; Blatt 10 ; Blatt 11 ; Blatt 12.
• Wintersemester 24/25: Hyperbolic Surfaces (2 SWS) ; Mirror Symmetry (2 SWS).

Humboldt-Universität zu Berlin

• Wintersemester 23/24: Exercice Sessions for Algebraic Geometry II. Lectures by Prof. Gavril Farkas.

Université Côte d'Azur

• 16-20 janvier 2023 : groupe Découverte de la période enjeux inter-semestre (niveau L1).
• du 15 novembre au 10 décembre 2022 : MathsFondements1 (niveau L1). La correction de l'interrogation 1 d'algèbre du 29 novembre.
• 18-22 janvier 2022 : groupe Découverte de la période enjeux inter-semestre (niveau L1). Voici le guide pour travailler le cours et les feuilles d'exercices distribuées au jour le jour : le guide de travail et planning ; lundi ; mardi ; mercredi ; jeudi ; vendredi.
• 1-11 septembre 2021 : module Maths0 de la période enjeux (niveau L1).

Au niveau prépa

• 9-13 juin 2025 : oral Math Ulm (filières MP et MPI). Voilà mes exercices et leurs corrigés. Merci à Armand Riera pour ses solutions alternatives !
• de septembre 2016 à mars 2020 : colles aux lycées Louis-le-Grand (MPSI3, MPSI4, MP*2) et Masséna (MP*).