Algebraic Geometry 1 [Fall 2019]
The prerequisites for this course are the standard undergraduate algebra courses on groups, rings and fields (see for example the course notes Algebra 1 and 2 on http://websites.math.leidenuniv.nl/algebra/ or chapter I-IV of S. Lang's 'Algebra', Springer GTM 211)
In particular, we assume that the student is familiar with the following notions: group, (commutative) ring, field, homomorphism, ideal, quotient ring, polynomial ring, domain, fraction field, principal ideal domain, unique factorization domain, topological space, continuous map, connected, Hausdorff, compact.
We will use the language of categories when useful, and with the necessary explanations. This material is reviewed during the Mastermath "Intensive course on Categories and Modules." This course is now online, we strongly recommend going through these videos (and the necessary written material there) in the week of xxx, or at least before the material is used in this course.
We will use some results from Commutative Algebra without giving proofs. It is recommended to follow the Commutative Algebra course that is offered on the same day at the same location, but this is not necessary.
Aim of the course
The course intends to give a first introduction to the basic notions and techniques of algebraic geometry.
We start by introducing the basic objects of algebraic geometry, namely algebraic varieties (affine, projective, general) and the morphisms between them. We then treat basic notions such as dimension, tangent space, differential 1-forms, smoothness. Some of the highlights of the course are the Riemann-Roch and Riemann-Hurwitz theorem. Applications of these theorems to curves in low genera are explored in the exercises.
Martijn Kool (Universiteit Utrecht, https://www.staff.science.uu.nl/~kool0009/)
Lenny Taelman (Universiteit van Amsterdam, https://staff.fnwi.uva.nl/l.d.j.taelman/)